Syracuse University
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.
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.
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.
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.
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 |
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.
`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`
`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.
`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).
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?).