The effect of a price of X change

Elly asked:

Hi Prof, can i ask you some questions regarding the budget line and indifference curve? When price of X falls, and X is normal, does it mean that normal good will always be on the right side of the budget line that has been separated by a point C? If that is the case, when price of X rises, and X is inferior, does the inferior good always falls on the right of budget line? Because from what i've known, when price of X rises, the budget line will rotate to the left from the original budget line, which means income decreases, so people will buy more inferior good, so inferior good will be on the right. Is that always true? I have also seen a few cases where the inferior good is on the left side even though Price of X rises and i cannot understand. Thank you for taking time to read my enquiries.

My response:

First, let's stick to the case where the price of X rises.  Afterward, the case where the price of X
falls can be worked through by doing the same analysis but in reverse.  Next, note that there are two effects to consider from a price change - a substitution effect and an income effect.  Let's consider those effects separately and then put them together.

Substitution effect

An increase in the price of X causes an increase in the relative price of X, because the price of Y has remained constant.  When a good's relative price has risen the substitution effect says less of the good (move to the left in the way Elly expresses it above).

Income effect

An increase in the price of X rotates the budget line inward around the Y intercept.  As long as some X was being consumed before the price change, that bundle is no longer affordable so this change means a reduction of real income.  The consequence of that income change on the amount of X consumed depends on whether X is normal or inferior. When X is normal the reduction of income leads to reduced consumption of X (again, that is a move to the left).  When X is inferior, the reduction of income leads to an increase in the consumption of X.

Overall

The substitution and income effects support each other when X is normal.  In this case the overall is to have less X consumed.  When X is inferior, however, the income effect offsets the substitution effect.  As an empirical matter we think that mainly the overall is determined by the substitution effect, so there still will be less X.  But it is logically possible for the income effect to win out, in which case the good is called a Giffen Good, named after the Scottish economist Sir Robert Giffen.