14.3 Decomposing the Effects of a Price Change

In this part of the course, we’ve been analyzing the effects of changes in the prices of goods, or in income. We’ve talked at length about how the optimal quantity changes; for example, we showed that if two goods are complements, a consumer will buy more of both or less of both in response to a change in the price of either. But we haven’t talked about why that’s the case.

One way of understanding the overall effect of a price change is to break it down into its component parts. An economist thinks about a price change in much the same way a physicist thinks about the path a cannonball takes through the air: the velocity vector $t$ seconds after being fired may be decomposed into its horizontal and vertical components, as shown in the diagram below.

[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/income_substitution/cannonball ]

(If you’re concerned about air resistance, click here for a more detailed explanation.)

Let’s think about one specific change: an increase in the price of good 2. (We could, of course, perform such a decomposition analyzing any kind of price change; the general approach would be the same.) When this occurs, there are two effects on a consumer’s budget set:

• Substitution effect: The slope of the budget set decreases, reflecting the fact that good 1 is now relatively less expensive than it was before: the higher price of good 2 means that the consumer doesn’t need to give up as much good 2 to get more good 1. This change in relative prices will generally cause the consumer to “substitute” good 1 for good 2 — that is, buy more good 1 and less good 2.
• Income effect: The size of the budget set decreases, reflecting the fact that the consumer can no longer afford all the bundles she used to be able to afford: that is, her real income (i.e. the value of her income in terms of the goods it can purchase) has decreased. As with any decrease in income, if both goods are normal goods (a term we’ll define precisely in the next lecture) this results in the consumer buying less of both goods.

In order to split the overall effect into these two effects, we need to make an important modeling choice. In particular, we would like the substitution effect to measure just the effect of the change in relative prices; and for the income effect to measure just the effect of the change in real income. We’ll do this by choosing an intermediate point, or “decomposition bundle,” that represents what the consumer would choose if the relative prices changed, but her “real income” was unaffected.

It’s important to note that there is no one decomposition bundle that we have to use; different textbooks use different methodologies. We’re going to follow the methodology known as the “Hicks decomposition,” after the Nobel Prize-winning British economist John Hicks. In this method, we will go through the following “thought experiment:”

• Jordan has income $m$.
• Initially, she faces some prices $(p_1,p_2)$; at those prices, she chooses some bundle $A$ and receives utility $U$.
• The price of good 2 rises to $p_2^\prime$. At this new price, if she still had income $m$, she would choose some bundle $C$ and receive utility $U^\prime$.
• Jordan’s mom cares about her, and wants her to be happy. She gives Jordan just enough money to be able to buy a bundle, $B$, that would give Jordan her initial utility of $U$.

Let’s see how this works. The following diagram shows the effect of a price increase from $p_2 = 2$ to $p_2 = 8$, holding $p_1 = 2$ constant. The overall result of this is to move from bundle $A$ to bundle $C$ This makes Jordan very unhappy, reducing her utility from $U_1$ (her initial indifference curve, passing through bundle $A$) to $U_2$ (her new indifference curve, passing through bundle $C$).

Now, you play the part of Jordan’s mom. You want to give her enough money to afford her initial utility $U_1$. Use the slider to shift $BL_2$ out until it reaches the initial indifference curve:

[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/income_substitution/hicks_decomposition_good2_compensation ]

As you can see, bundle B occurs at a special point: it’s the point along Jordan’s initial indifference curve where her MRS is equal to the new price ratio. One way of thinking about this is that it’s the lowest-cost way of achieving her initial utility given the new prices. We call this bundle the Hicks decomposition bundle.

Income and substitution effects

Having derived the Hicks bundle $B$, we can use it to decompose the overall effect of the price change $(A \rightarrow C)$ into the substitution effect ($A \rightarrow B$) and the income effect ($B \rightarrow C$):

[ See interactive graph online at https://www.econgraphs.org/graphs/consumer/income_substitution/hicks_inc_sub_effect_good2 ]

Recall that the substitution effect analyzes the effect of the change in relative prices. Specifically, it moves along the original indifference curve, from point $A$ (where the MRS equals the original price ratio) to point $B$ (where the MRS equals the new price ratio).

Next, recall that the income effect analyzes the effect of the change in purchasing power. Note that the new budget line $BL_2$ and the compensated budget line have the same slope – that is, the new price ratio – but different levels of income. We can think of the difference in those levels of income (i.e. the amount you had to compensate her in the exercise above) as representing the change in her purchasing power. Thus, the income effect causes her to move from bundle $B$ to bundle $C$.