The demand for a luxury good whose purchase would exhaust a big portion of one's income is

Relatively price elastic.

Perfectly price inelastic.

Perfectly price elastic.

Relatively price inelastic.

Relatively price elastic

Explanation:

Price elasticity is defined as the percentage change in quantality demanded with respect to percentage change in price. Lets explore it in detail.

Goods whose quantity demanded remains stable irrespective of the change in price (in percentage terms), have perfectly price inelastic demand. Goods whose quantity demanded changes less (in percentage terms) then the change in their price (in percentage terms), have relatively price inelastic demand. Goods whose quantity demanded disappears with even a negligible change in price (in percentage terms), have perfectly price elastic demand. Goods whose quantity demanded changes more (in percentage terms) then the change in their price (in percentage terms), are relatively price elastic.

Perfectly inelastic goods are the ones which have no substitute for example oxygen. If firms start selling us oxygen, we will have no other option except to buy it. Perfectly elastic goods have lots of substitute. So if the price of a product increases with minimal amount, the consumer will shift to its substitutes.

Items that are considered necessity like bread are relatively price inelastic. Luxury goods, on the other hand, have relatively price elastic demand. Let's see an example.

Bread is a necessity as it is a staple food. If the price of bread increases from $1 to $2 (100% increase), the consumption will not drop significantly. Instead a person having three meals a day, might come to two meals (33% decrease). Hence demand for bread is relatively price inelastic.

As mentioned in the question, demand for luxury goods whose purchase would exhaust a big portion of one's income is relatively price elastic. Since it takes a large share of wallet, a slight increase in price will discourage the consumer to spend on it at all (it's not a need so why bother spending).