9.3 Hicks Decomposition

Now that we can do cost minimization, we have a way of approaching the Hicks decomposition.

The following diagram shows the effect of an increase in the price of good 1, holding $p_2$ and $m$ constant.

Let’s break down what’s going on in this diagram.

• Utility maximization with initial prices: Budget line $BL_1$ represents the initial budget line (before the price change); if faced with that budget constraint, Jordan would choose bundle $A$ and achieve utility $U_1$. This bundle occurs at the point where the initial budget line $BL_1$ intersects the original income offer curve $IOC_1$.
• Utility maximization with final prices: Budget line $BL_2$ represents the final budget line (after the price change); if faced with that budget constraint, Jordan would choose bundle $C$ and achieve utility $U_2$. This bundle occurs at the point where the initial budget line $BL_2$ intersects the final income offer curve $IOC_2$.
• Cost minimization with initial utility and final prices: Bundle $B$ represents the “Hicks decomposition point” between bundles A and C: specifically, it shows the bundle that minimizes the cost of achieving the Jordan’s initial utility when facing the new prices. This bundle occurs at the point where the initial indifference curve $U_1$ crosses the final income offer curve $IOC_2$.

In the diagram above, you can check the “show income offer curves” box to see the income offer curves before and after the price change, and the “show price offer curve for good 1” to see that as well. Some important things to note:

• Bundles $A$ and $C$ lie along the same price offer curve. The movement from $A$ to $C$ represents the “total effect” of the price change. In this case, the POC is upward sloping, indicating that these two goods are complements.
• Bundles $B$ and $C$ lie along the same income offer curve. This is because the movement from $B$ to $C$ represents the consumer’s change in real income, holding prices constant. Put another way: if Jordan’s mom actually did compensate her enough to keep her utility at $U_1$, the movement from $C$ to $B$ along $IOC_2$ would represent the increase in income.