IAI Oct 2006 Q30

'Linearly homogeneous' isn't a phrase I've seen in the IFoA's current CB2 course, but looking at the IAI's indicative solution I think it means that if we halve the consumption of a commodity then utility from that commodity halves.

Assuming that is true, my interpretation of the IAI solution is that:
- an individual will have a certain budget or amount of money they can spend
- at present an individual will be purchasing a combination of these two commodities that maximises their utility, given their budget
- if the price of the commodities double, then the overall amount of the goods that an individual can afford to purchase will fall. In fact the amount will halve, unless they change the ratio in which they consume the goods eg reduce consumption of one good by more than 50% so they can reduce consumption of the other good by less than 50%
- as utility derived from each of these goods also halves, the individual is not tempted to change the ratio in which they consume the goods (as the current ratio is maximising utility)
- so the individual halves the consumption of each good

I hope that helps.