Econ 50Q Section 8: Measuring Welfare Effects of a Price Change

Compensating Variation

So far, we have spent much of our time thinking about how a consumer responds to a price change, but we have spent much less time on understanding how good or bad that change for them. Increasing prices are generally bad, and decreasing prices are generally good, but can we get more rigorous in quantifying these effects? A lot of this sort of analysis, usually called Welfare Analysis, comes down to translating between the language of money and the language of utility.

Compensating variation is one way to make this relationship precise. Following a price change, we imagine compensating the consumer just enough to afford their initial utility at the new prices. Compensating variation asks the following: how much extra cash would the consumer need to receive to be just as happy as they were before?

To illustrate, let’s think of a consumer with the utility function \(u(x_1,x_2) = (x_1 \times x_2)^{1 \over 2}\) who starts with $m = 8$ dollars, and initially face prices $p_1 = p_2 = 1$. What happens when 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 compensated 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. We can find the bundle that achieves this by solving the cost minimization problem: it is just the Hicksian (compensated) bundle for the initial utility at the new prices. Here that bundle is $(2,8)$; 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.

Recall 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:

Equivalent Variation

Like compensating variation, equivalent variation is an estimate, in dollar terms, of the welfare effect of a price change. However, while compensating variation measures the amount of income a consumer would need to be as happy as they were before the price change, equivalent variation asks: if the prices hadn’t changed, what change in income would have resulted in the same change in utility?

As before, let’s experiment with this and then derive it mathematically. The graph shows the same price change as we were just examining, which reduces the consumer’s utility from $U = 4$ to $U^\prime = 2$. Here, instead of compensating them for this loss of utility, we’re trying to figure out how much of a loss of income would have resulted in the same utility loss if the prices had not changed. Use the slider to take money away from them until their utility drops to 2:

As you should see, if you reduce the consumer’s income by $€4$, they would have a total of $€4$. With that income, their utility-maximizing bundle would be $(2,2)$, and they would have a utility of $u(2,2) = (2 \times 2)^{1 \over 2} = 2$, which is their final utility after the price change. Therefore their equivalent variation of this price change is this reduction of $€4$.

As with CV, there are two ways to compute this equivalent variation:

Method 1: Cost minimization

This is the most bulletproof method, and the one I recommend on exams. Here we’re trying to figure out the cost-minimizing way of achieving the new utility at the initial prices. Since the initial prices were $p_1 = p_2 = 1$, this means minimizing the expenditure $x_1 + x_2$ subject to the utility constraint $(x_1 \times x_2)^{1 \over 2} = 2$; this results in the bundle $(2,2)$. At the original prices, this bundle would cost $€4$; since the consumer has $m = 8$, the equivalent variation is $|8 - 4| = 4$.

Method 2: Expenditure

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

Recall 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 $p_1 = p_2 = 1$, the amount of money required to achieve $U = 2$ is \(E(1,1,2) = 2 \times 1^{1 \over 2} \times 1^{1 \over 2} \times 2 = 4\) as we found before.

Reading Quiz

That's it for today! Click here to take the quiz on this reading.