Using Slutsky method to understand substitution and income effect for luxury goods

Question

If both goods are luxury goods, such as Channel vs Dior bags, how can you graph the substitution and income effect using the Slutsky method when the price of Channel bags fall?

What I've tried:

1. Channel on the x-axis and Dior on the y-axis.
2. Tangent budget constraint (BC1) to utility curve (U1) intersects at optimal point (A).
3. If price of Channel falls, then the demand will decrease (given that it is a luxury good). This make the new budget constraint steeper (BC2). Is the substitution effect the decrease of Channel from point A to B?
4. I'm not sure how to draw the third budget constraint to reflect the income change. Would it shift outwards or inwards? I'm assuming it would have to shift outwards, but I'm not sure where it would intersect on the y-axis.

Answer 1

I think that you are confusing a luxury good with an inferior good.

First, a luxury good is a superior good, and a superior good is NORMAL (which means that the demand of that good increases as consumer's wealth rises), and for a superior good the income elasticity is greater than one (which means, if income increases by $1 then the demand of the good increases more than 1 unit).

Note, that the substitution effect(SE) moves ALWAYS in the opposite direction to change in prices regardless of the nature of the good (even if it is inferior). This occurs because the SE is captured by the hicksian demand function, which satisfies the compensated law of the demand. The hicksian demand simply gives you the consumption bundles minimizing expenditures for a fixed level of utility and at a different price levels. So, if the price of one commodity goes down, the Hicksian demand for that commodity always increases (even if the good is inferior), because the substituion effect simply tells us which is the change in demand in commodities to stay on the initial indifference curve and simultanously minimizing expenditures. If a good becomes cheaper because its price has dropped, is natural to consumer more of that good and less of the other, in a way that you are on a lower isoexpenditure line but still on the initial indifference curve.

To answer to your question, treat the luxury good as a NORMAL good, with a particularly strong income effect. So, if the price of $x_1$ goes down, the budget line becomes flatter. You will find an isoexpenditure line parallel to the new budget line, a positive SE, and an even positive income effect which renforces the SE. From the Slutsky equation, you can infer the slope of the Walrasian demand of your Chanel bag, and see that is negative, because you always have $\frac{\partial h_j (p,u)}{\partial p_j} \le 0$, where $h_j(p,u)$ is your Hicksian demand, and since the good is normal, $\frac{\partial x_j (p,w)}{\partial w} > 0$, which in the Slutsky equation, when you look at the slope of the Walrasian demand, you have a minus in front of this partial derivative.