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Compensating Variation

Compensating variation can be used to calculate the effect of a price change on an individual's overall welfare. The best way to understand it is to work through it graphically and go through a numerical example.

Graphical Analysis

Suppose we're interested in the effect of an increase in the price of good X on a particular person. The price goes from Px1 to Px2, where Px2 is greater than Px1. The individual has M dollars to allocate between X and Y, where Y represents all other goods and has price Py. Py and M do not have a subscripts because they will remain constant throughout the analysis.

Step 1: Find the Initial Equilibrium

Her budget constraint before the price increase is the following:
BC1: M = Px1*X1 + Py*Y1

Given the budget constraint, she will choose bundle 1 in the diagram below and will be on indifference curve IC1.


Step 2: Find the New Equilibrium Following Price Increase

When the price of X rises to Px2, the budget constraint changes to the following:
BC2: M = Px2*X2 + Py*Y2

The new constraint is labeled BC2 in the diagram. She now chooses bundle 2 and is clearly worse off because IC2 is lower than IC1.


Step 3: Finding the Compensating Variation

Now imagine giving her some extra money to spend after the price change. We could, in principle, give her just enough extra money for her to be able to reach her original indifference curve, IC1, at bundle 3.

If we did this, she would be just as well off as she was at the beginning -- the extra income would compensate her for the price increase. Using CV to represent the extra money, the budget constraint becomes:

BC3: M + CV = Px2*X3 + Py*Y3

Numerical Example

To calculate the numerical value of a compensating variation we need to use the individual's utility function. Suppose the person above has a Cobb-Douglas utility function of the following form:

`U = X^0.3 * Y^0.7`

It can be shown that her demands for X and Y will be:

`X = (0.3M)/P_x`

`Y = (0.7M)/P_y`

The utility function and demand equations are the basis for the compensating variation calculation. Here's how it works.

Step 1: Find the Initial Utility

Suppose that M is $1000, Px1 is $2 and Py is $5. Using her demand equations to calculate the amount of X and Y she buys (the components of bundle 1):

`X_1 = (0.3*$1000)/($2) = 150`

`Y_1 = (0.7*$1000)/($5) = 140`

From this, we can calculate her original utility, U1. That's crucial because the object of the compensating variation calculation is to figure out how much money it would take to return her to IC1, and hence to U1. Here's the calculation:

`U_1 = X_1^0.3 * Y_1^0.7`

`U_1 = 150^0.3 * 140^0.7 = 142.93`

Step 2: Derive the Expenditure Function

The next step requires a bit of algebra. The good news is that it only needs to be done once for each utility function. First, insert the demand equations into the utility function:

`U = X^0.3 * Y^0.7`

`U = ((0.3*M)/P_x)^0.3 * ((0.7*M)/P_y)^0.7`

Next, factor out the M terms and collect them together:

`U = (0.3/P_x)^0.3  * M^0.3 * (0.7/P_y)^0.7 * M^0.7`

`U = (0.3/P_x)^0.3 * (0.7/P_y)^0.7 * M`

The equation simplified a lot because of a property of exponents: `A^b*A^c=A^(b+c)`. In this case:

`M^0.3 * M^0.7 = M^(0.3+0.7) = M`

Now, rearrange the equation to solve for M in terms of the other variables:

`M = U /( (0.3/P_x)^0.3 * (0.7/P_y)^0.7)`

`M = U * (P_x/0.3)^0.3 * (P_y/0.7)^0.7`

The last expression is known as the "expenditure function" corresponding to her utility function. It is the key to the compensating variation calculation.

Step 3: Apply the Expenditure Function

The great thing about the expenditure function is that it lets us calculate how much money she'd need to get any particular utility at any set of prices. For example, as a check we could calculate how much money she'd need to get U1 at Px1 and Py. The result had better be $1000 since that's how much she had when we determined U1 in the first place. Going through the calculation:

`M_1 = U_1 * (P_{x1}/0.3)^0.3 * (P_y/0.7)^0.7`

`M_1 = 142.93 * ({$2}/0.3)^0.3 * ({$5}/0.7)^0.7 = $1000`

That's good: the equation correctly generates her original M. We can now use the expenditure function to calculate how much money she would need in order to reach bundle 3.

Suppose the price of X, Px2, has risen to $4. To figure out how much money she would need to get back to IC1, which will be at point 3, we can plug U1 and Px2 into the expenditure function:

`M_3 = U_1 * (P_{x2}/0.3)^0.3 * (P_y/0.7)^0.7`

Notice that the only change from our previous calculation is that Px1 has been replaced by the new price, Px2. Inserting the corresponding numbers and doing the calculation:

`M_3 = 142.93 * ({$4}/0.3)^0.3 * ({$5}/0.7)^0.7 = $1231`

That says that she would need $1231 dollars in order to be able to reach IC1 after the price of X has risen (that is, M would have to have been $1231).

Step 4: Calculating the Compensating Variation

The hard work is done. To calculate the compensating variation, we just subtract her actual M from the value calculated in the previous step. Since she would need $1231 to reach IC1, but only had $1000, the amount that would compensate her for the price change is $231.

An Important Fact to Note

Compensating variation is the amount of additional money someone would need to reach her initial utility, not her initial consumption bundle. In general, the CV will be smaller than the amount of money she'd need to buy her original consumption bundle. Both the graphical and numerical examples illustrate this point. The CV returns her to IC1, not to bundle 1.

The only time the CV is large enough for the individual to be able to buy her original bundle occus when she regards X and Y as perfect complements (can you see why?).

Additional Information

Exercises on Compensating Variation
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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 10/24/2018