# Income and substitution effect for perfect substitutes

I was recently asked about what the income and substitution effects are for perfect substitutes are. Given the rather peicewise nature of the demands for each good in a utility function considering perfect substitutes I'm not sure what the answer is.

That being said, what are the income and substitution effects for a utility function considering goods that are perfect substitutes?

An indifference curve for perfect substitutes is a straight line. In fact it is the line defined by $$y=const-x$$, for a utility level of $$const\in\Bbb R$$. We maximize the utility when our budget line is tangent to the IC line. But they are both straight lines, so there are a few cases (considering a situation with only 2 goods):

1. the prices are not equal ($$P_X< P_Y$$ for example, then you would always choose good $$X$$)
2. $$P_X> P_Y$$ for example, then you would always choose good $$Y$$
3. the prices of the goods are equal $$\implies$$ our budget line is parallel to the IC line and the utility maximization point is anywhere on the budget line.

For cases 1 and 2:

Let's consider the case where good $$Y$$ is cheaper and becomes relatively more expensive after a price change.

We can use Hicks' method to determine the SE and the IC by reducing artificially the consumer's income so that he ends up consuming a basket that is on the original IC curve before the price change (I think Slutksy method is impossible to implement, because we would have to reduce the budget so that the original basket is chosen but with the new prices it is impossible, unless the good is an inferior good I guess, but can they are perfect substitutes so they are both inferior, which I'm not sure if it's possible because an inferior good must take up most of your budget)

We go from equilibrium $$E$$ to $$E'$$ and the equilibrium $$E''$$ is the one due to a reduced budget by the method of Hicks.

For the third case, in theory the original equilibrium can be anywhere on the budget line. In practice it is probably in a fixed position and you can use the same method to derive the IE and the SE.