10.3 Compensating Variation

When we first introduced the Hicks Decomposition, we motivated it as a thought experiment: what if, following a price change, we “compensated” the consumer just enough to afford their initial utility at the new prices. The concept of compensating variation just asks: how much would the consumer need to be compensated?

To illustrate, let’s continue to think of a consumer with the utility function \(u(x_1,x_2) = (x_1 \times x_2)^{1 \over 2}\) Let’s suppose they have $m = 8$ dollars, and initially face prices $p_1 = p_2 = 1$; but then the price of good 1 increases to $p_1^\prime = 4$. As the diagram below illustrates, before the price change they would choose bundle $A = (4,4)$ and receive utility $U = u(4,4) = (4 \times 4)^{1 \over 2} = 4$; and after the price change they would choose bundle $C = (1,4)$ and receive utility $U^\prime = u(1,4) = (1 \times 4)^{1 \over 2} = 2$.

The question before us is: how much additional money do we need to give this person in order for them to get back to their initial indifference curve? Use the slider in the graph below to “compensate” them:

As you should see, if you give the consumer an additional $€8$, they would have a total of $€16$. With that income, their utility-maximizing bundle would be $(2,8)$, and they would have a utility of $u(2,8) = (2 \times 8)^{1 \over 2} = 4$, which was their initial utility before the price change. Therefore their compensating variation of this price change is this additional $€8$ that they need to be “made whole” after the price increase.

There are two ways to compute this compensating variation:

Method 1: Evaluate the cost of the Hicks decomposition bundle

This is the most bulletproof method, and the one I recommend on exams. Fundamentally, what we’re doing here is trying find the cheapest way to afford the initial utility at the new prices. Following the same Hicks decomposition procedure as in the last chapter, we can find the bundle that achieves this by solving the cost minimization problem, then evaluating how much that bundle costs at the new prices. In this case, the Hicks decomposition process will find the bundle $(2,8)$; at $p_1^\prime = 4$ and $p_2 = 1$, that bundle would cost $p_1^\prime x_1 + p_2x_2 = 4 \times 2 + 1 \times 8 = 16$; since the consumer already has $m = 8$, the compensating variation is $|16 - 8| = 8$.

Method 2: Use the expenditure function

This is the more elegant method. The expenditure function describes how much a given amount of utility costs at any prices. Therefore, you can find the cost of the initial utility at the new prices by plugging those numbers into the expenditure function.

In the last section we established that the expenditure function for this utility function is \(E(p_1,p_2,U) = 2p_1^{1 \over 2}p_2^{1 \over 2}U\) Therefore, when $p_1^\prime = 4$ and $p_2 = 1$, the amount of money required to achieve $U = 4$ is \(E(4,1,4) = 2 \times 4^{1 \over 2} \times 1^{1 \over 2} \times 4 = 16\) as we found before.

Note that we can also use the expenditure function to interpret the compensating variation visually. As we said before, when $p_1 = p_2 = 1$, the equation of the expenditure function is $E(U) = 2U$. When $p_1$ increases to 4, every level of utility becomes more expensive: the expenditure function becomes $E(U) = 4U$. Therefore, the amount of money required to achieve $U = 4$ increases from 8 to 16: