I am trying to figure out the following question involving the Paasche price index:
- two goods are perfect substitutes (let's assume for simplicity that indifference curve has slope of -1)
- relative price of good 1 to good 2 is 1/2 in country A and I am moving to country B
Question: In which scenario does a Paasche price index cost of living adjustment to my salary in moving from country A to B leave me worse off?
I am confused because it seems to me that with perfect substitutes the Paasche index is the true index but there must be a scenario where that is not the case..
If I understand the definitions right, adjusting your income according to the Paasche index means that if you end up with spending your salary on a certain basket of goods in the new country, exactly the same basket of goods would cost you your old salary in your old country. This never makes your life better because, assuming that you always optimize your basket, the new basket of goods that is an optimum in country B would be already affordable in country A, but nobody said that you might not be able to optimize further while in A. It is a fair game though, if the new basket is also optimal in A.
In your two perfect substitutes scenario, the optimal basket just consists of a single good (the cheaper one). In country A, it is good 1, so if it remains good 1 in country B, the game will be fair: the Paasche index gives you the coefficient $\frac {p_B(1)q_B(1)+p_B(2)q_B(2)}{p_A(1)q_B(1)+p_A(2)q_B(2)}=\frac {p_B(1)}{p_A(1)}$ because $q_B(2)=0$ (you still ignore good 2) and your salary is adjusted according to the cost of good 1. However, as soon as the price $p_B(1)$ of good 1 in the country B is greater than the price $p_B(2)$ of good 2, you'll switch your basket to pure good 2 when moving from A to B, and the result will be that the Paasche index will be the ratio of prices of good 2, i.e., you'll end up twice worse off because in country A you could derive twice as much utility as you can afford now in country B on your new salary by using good 1 instead.