9 Slutsky Equation – A Microeconomic Theory Workbook

9.1 Objective

In this chapter, I’ll help you understand how to think like an economist about the sometimes ambiguous effects of a price change. For instance, when people are offered higher wages, we typically expect them to work more in response, like Uber drivers working extra during surge pricing. But what if your wage increased from $10 an hour to $1000 an hour? Would you really work more? Or might you decide to work fewer hours and use some of the money you’ve earned to do other things? What if your wage was $1,000,000 an hour? Wouldn’t you work less? In this chapter, we’ll sort out these effects.

9.2 Income and Substitution Effects

When the price of a good changes, two main effects occur: the rate at which you can exchange one good for another changes (slope of the budget line), and the total purchasing power of your income is altered (budget line is shifted).

If good 1 becomes cheaper, you need to give up less of good 2 to purchase good 1. The change in demand due to the altered rate of exchange between the two goods is called the substitution effect.

Additionally, if good 1 becomes cheaper, your money income can now buy more of good 1. This increase in your purchasing power leads to a change in demand, known as the income effect.

9.3 The Sign of the Substitution Effect

The sign of the income effect can be positive or negative, depending on whether the good is normal or inferior.

What about the sign of the substitution effect? In this section, I’ll make the argument that if the price of a good falls, the substitution effect must be to consume more of that good.

The image above outlines the proof: $X$ is directly revealed preferred to all bundles in the blue area. Because $X$ is still affordable under the substitution effect budget line, no bundle in the blue area will be chosen, specifically no bundle along the pink section of the substitution effect budget line. The section that’s left over is the yellow section, where the consumer chooses more $x_1$ than the original bundle. Therefore, when $p_1$ falls, the substitution effect is always that the consumer will choose more of $x_1$.

9.4 Slutsky Equation

The Slutsky Equation just says that, given a change to $p_1$, the total change in demand for good 1 is equal to the change in demand for good 1 from the substitution effect plus the change in demand for good 1 from the income effect:

\[\Delta x_1 = \Delta x_1^S + \Delta x_1^I\]

Suppose $p_1$ falls. In the previous section, I showed why $\Delta x_1^S \geq 0$. If $x_1$ is a normal good, then $\Delta x_1^I$ will also be greater than 0, and so the demand for good 1 is a positive number plus a positive number: it will certainly increase.

If good 1 is inferior though, $\Delta x_1^I \leq 0$. If that income effect is more negative than the substitution effect is positive, $\Delta x_1$ may be negative in total. That means that when the price of the good falls, demand for it decreases: it’s a Giffen good. This is why Giffen goods must always be inferior goods.

The Slutsky Equation lets us see that demand for Inferior goods may have ambiguous price effects, but the demand for Normal goods do not. The Law of Demand follows from that fact:

Law of Demand: If the demand for a good increases when income increases (the good is a normal good), then the demand for that good must decrease when its price increases.

9.5 Extra Credit Opportunity: Reviewing Quiz 4 Part B

9.6 Classwork 9

Suppose a consumer’s income $m = 10$, $p_1 = 2$, and the consumer’s demand is given by $x_1 = 2 + \frac{m}{5 p_1}$. Let $p_2$ represent the composite good. If $p_1$ fell to $1$, what is the change to $x_1$ and how much of that change is the substitution effect versus the income effect?
Let $x_1$ and $x_2$ be perfect complements, like right and left shoes. Recall that indifference curves are L shaped and utility is given by $u(x_1, x_2) = \min (x_1, x_2)$.
- The demand for goods 1 and 2 can be derived from these two equations ($x_1$ and $x_2$ are the 2 unknowns): the first equation is that you want the same number of right and left shoes, so $x_1 = x_2$, and the second equation is that the budget constraint holds with equality: $p_1 x_1 + p_2 x_2 = m$. Solve for $x_1$ to show that for perfect complements, a consumer acts as if goods 1 and 2 are a single good with a price $p_1 + p_2$.
- If you want to spend $6 on shoes, right shoes cost $1, and left shoes cost $2, how many right shoes will you buy? How many left shoes?
- Take the conditions from the previous problem. If the price of left shoes went down to $1, how many right and left shoes would you buy now, and how many more from the substitution effect versus the income effect?