# The Slutsky Equation Revisited

The above applications of revealed preference are handy, but they don't really answer the main question: how does the demand for a good react to a change in its price? We saw in Chapter 8 that if money income was held constant, and the good was a normal good, then a reduction in its price must lead to an increase in demand.

The catch is the phrase "money income was held constant." The case we are examining here necessarily involves a change in money income, since the value of the endowment will necessarily change when a price changes. A Net demand 6 Gross demand C Net supply

Gross demand, net demand, and net supply. Using the gross demand and net demand to depict the demand and supply behavior.

In Chapter 8 we described the Slutsky equation that decomposed the change in demand due to a price change into a substitution effect and an income effect. The income effect was due to the change in purchasing power when prices change. But now, purchasing power has two reasons to change when a price changes. The first is the one involved in the definition of the Slutsky equation: when a price falls, for example, you can buy just as much of a good as you were consuming before and have some extra money left over. Let us refer to this as the ordinary income effect. But the second effect is new. When the price of a good changes, it changes the value of your endowment and thus changes your money income. For example, if you are a net supplier of a good, then a fall in its price will reduce your money income directly since you won't be able to sell your endowment for as much money as you could before. We will have the same effects that we had before, plus an extra income effect from the influence of the prices on the value of the endowment bundle. We'll call this the endowment income effect.

In the earlier form of the Slutsky equation, the amount of money income you had was fixed. Now we have to worry about how your money income changes as the value of your endowment changes. Thus, when we calculate the effect of a change in price on demand, the Slutsky equation will take the form:

total change in demand = change due to substitution effect + change in demand due to ordinary income effect + change in demand due to endowment income effect.

The first two effects are familiar. As before, let us use Ax\ to stand for the total change in demand, Ax* to stand for the change in demand due to the substitution effect, and Ax™ to stand for the change in demand due to the ordinary income effect. Then we can substitute these terms into the above "verbal equation" to get the Slutsky equation in terms of rates of change:

Axi Axs Axm

—— = -—- — x\ A 1 + endowment income effect. (9.1)

### Api Api Am

What will the last term look like? We'll derive an explicit expression below, but let us first think about what is involved. When the price of the endowment changes, money income will change, and this change in money income will induce a change in demand. Thus the endowment income effect will consist of two terms:

endowment income effect = change in demand when income changes x the change in income when price changes. (9.2)

Let's look at the second effect first. Since income is defined to be m = piu)i we have

### Am Api

This tells us how money income changes when the price of good 1 changes: if you have 10 units of good 1 to sell, and its price goes up by \$1, your money income will go up by \$10.

The first term in equation (9.2) is just how demand changes when income changes. We already have an expression for this: it is Ax™/ Am: the change in demand divided by the change in income. Thus the endowment income effect is given by

Axm Am Ax171 endowment income effect = . 1 -— = A 1 cji . (9.3)

Am Api Am v

Inserting equation (9.3) into equation (9.1) we get the final form of the Slutsky equation: 