10.4 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.
As before, we’ve 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 $p_1 = p_2 = 1$, 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.