The theory of consumer's surplus is very tidy in the case of quasilinear utility. Even if utility is not quasilinear, consumer's surplus may still be a reasonable measure of consumer's welfare in many applications. Usually the errors in measuring demand curves outweigh the approximation errors from using consumer's surplus.
But it may be that for some applications an approximation may not be good enough. In this section we'll outline a way to measure "utility changes" without using consumer's surplus. There are really two separate issues involved. The first has to do with how to estimate utility when we can observe a number of consumer choices. The second has to do with how we can measure utility in monetary units.
We've already investigated the estimation problem. We gave an example of how to estimate a Cobb-Douglas utility function in Chapter 6. In that example we noticed that expenditure shares were relatively constant and that we could use the average expenditure share as estimates of the Cobb-Douglas parameters. If the demand behavior didn't exhibit this particular feature, we would have to choose a more complicated utility function, but the principle would be just the same: if we have enough observations on demand behavior and that behavior is consistent with maximizing something, then we will generally be able to estimate the function that is being maximized.
Once we have an estimate of the utility function that describes some observed choice behavior we can use this function to evaluate the impact of proposed changes in prices and consumption levels. At the most fundamental level of analysis, this is the best we can hope for. All that matters are the consumer's preferences; any utility function that describes the consumer's preferences is as good as any other.
However, in some applications it may be convenient to use certain monetary measures of utility. For example, we could ask how much money we would have to give a consumer to compensate him for a change in his consumption patterns. A measure of this type essentially measures a change in utility, but it measures it in monetary units. What are convenient ways to do this?
Suppose that we consider the situation depicted in Figure 14.4. Here the consumer initially faces some prices 1) and consumes some bundle (x\, #2). The price of good 1 then increases from p{ to pi, and the consumer changes his consumption to (x\, x2). How much does this price change hurt the consumer?
The compensating and the equivalent variations. Panel A shows the compensating variation (CV), and panel B shows the equivalent variation (EV).
One way to answer this question is to ask how much money we would have to give the consumer after the price change to make him just as well off as he was before the price change. In terms of the diagram, we ask how far up we would have to shift the new budget line to make it tangent to the indifference curve that passes through the original consumption point (a?!,^). The change in income necessary to restore the consumer to his original indifference curve is called the compensating variation in income, since it is the change in income that will just compensate the consumer for the price change. The compensating variation measures how much extra money the government would have to give the consumer if it wanted to exactly compensate the consumer for the price change.
Another way to measure the impact of a price change in monetary terms is to ask how much money would have to be taken away from the consumer
before the price change to leave him as well off as he would be after the price change. This is called the equivalent variation in income since it is the income change that is equivalent to the price change in terms of the change in utility. In Figure 14.4 we ask how far down we must shift the original budget line to just touch the indifference curve that passes through the new consumption bundle. The equivalent variation measures the maximum amount of income that the consumer would be willing to pay to avoid the price change.
In general the amount of money that the consumer would be willing to pay to avoid a price change would be different from the amount of money that the consumer would have to be paid to compensate him for a price change. After all, at different sets of prices a dollar is worth a different amount to a consumer since it will purchase different amounts of consumption.
In geometric terms, the compensating and equivalent variations are just two different ways to measure "how far apart" two indifference curves are. In each case we are measuring the distance between two indifference curves by seeing how far apart their tangent lines are. In general this measure of distance will depend on the slope of the tangent lines—that is, on the prices that we choose to determine the budget lines.
However, the compensating and equivalent variation are the same in one important case—the case of quasilinear utility. In this case the indifference curves are parallel, so the distance between any two indifference curves is the same no matter where it is measured, as depicted in Figure 14.5. In the case of quasilinear utility the compensating variation, the equivalent variation, and the change in consumer's surplus all give the same measure of the monetary value of a price change.
EXAMPLE: Compensating and Equivalent Variations
Suppose that a consumer has a utility function u(x i,x2) = x\x2- He originally faces prices (1,1) and has income 100. Then the price of good 1 increases to 2. What are the compensating and equivalent variations?
We know that the demand functions for this Cobb-Douglas utility function are given by m
2pi m
Using this formula, we see that the consumer's demands change from {x\,xl) = (50,50) to (xux2) = (25,50).
To calculate the compensating variation we ask how much money would be necessary at prices (2,1) to make the consumer as well off as he was consuming the bundle (50,50)? If the prices were (2,1) and the consumer
Quasilinear preferences. With quasilinear preferences, the distance between two indifference curves is independent of the slope of the budget lines.
had income m, we can substitute into the demand functions to find that the consumer would optimally choose the bundle (m/4,m/2). Setting the utility of this bundle equal to the utility of the bundle (50, 50) we have
Hence the consumer would need about 141 —100 = $41 of additional money after the price change to make him as well off as he was before the price change.
In order to calculate the equivalent variation we ask how much money would be necessary at the prices (1,1) to make the consumer as well off as he would be consuming the bundle (25,50). Letting m stand for this amount of money and following the same logic as before,
Thus if the consumer had an income of $70 at the original prices, he would be just as well off as he would be facing the new prices and having an income of $100. The equivalent variation in income is therefore about 100 - 70 - $30.
EXAMPLE: Compensating and Equivalent Variation for Quasilinear Preferences
Suppose that the consumer has a quasilinear utility function + x2.
We know that in this case the demand for good 1 will depend only on the price of good 1, so we write it as x\(p\). Suppose that the price changes from pi to pi. What are the compensating and equivalent variations?
At the price p\, the consumer chooses = xi(pj) and has a utility of v{x*) + m — p\x\. At the price p1, the consumer choose x\ = ^i(pi) and has a utility of v(x\) + rn — p\X\.
Let C be the compensating variation. This is the amount of extra money the consumer would need after the price change to make him as well off as he would be before the price change. Setting these utilities equal we have v(xi) -f m + C — pixi = + m — Pi^i.
Solving for C we have
Let E be the equivalent variation. This is the amount of money that you could take away from the consumer before the price change that would leave him with the same utility that he would have after the price change. Thus it satisfies the equation v(xl) + m — E — p\x\ ~ v(x\) + m — p\x\.
Solving for E, we have
Note that for the case of quasilinear utility the compensating and equivalent variation are the same. Furthermore, they are both equal to the change in (net) consumer's surplus:
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