#### True Cost-of-Living Adjustments

We may now ask how large the necessary compensation is that leaves our consumers as well off in the second period as in the first? In the example above we used the following utility function: U(C, F) = C0.5F0.5. The initial utility was U1 = 50. Let’s denote the unknown income Y ∗, the demand functions for F and C are:

F2(pF , Y) = (0.5 x Y)/PF , and C2(pC, Y) = (0.5 x Y)/PC,

and the utility is:

U(F2,C2) = [(0.5 x Y)/PF]0.5 [(0.5 x Y)/PC]0.5 = U1
= (0.5/PF)0.5 (0.5/PC)0.5  Y = U1

Hence, we can solve for Y as:

Y= U1 [ (P2F/0.5)0.5 (P2C/0.5)0.5]

Now we put in the numbers for the prices and the initial utility level:

Y = 50 (1.03/0.5)0.5 (1/0.5)0.5 = 101.49.

The difference is not big, but the minimum income increase necessary to exactly compensate the consumers is less than the increase in the CPI. The “true” cost-of-living index in this case is Y/Y1 which is less than the CPI.
The CPI calculates the increase in prices as: Y2/Y1, which we can write as:

Y2/Y1 = (Y/Y1) . (Y2/Y)

The first term on the right-hand side shows the increase in the true cost-of-living and the second term, Y2/Y, reflects the substitution bias in the CPI. This terms if greater than one because Y2 > Y, hence the bias is positive. There is no substitution bias if all prices increase at the same rate so that relative prices remain constant. The faster some prices rise relative to others the more pronounced is the upward bias due to the substitution effects.

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