Two persons, A and B, are dining together. Their utility functions for food are respectively expressed by 5 log (xA) and by 7 log (xB), where xA and xB are the quantities of food, measured in suitable, continuous units. The net utility is then calculated by subtracting the sum paid by each individual i = A, B from the
individual's utility from consumption. The unit price of food is equal to 1.
3.1.1. What would be the quantity of food ordered by each individual, if each one of them were separately billed?
3.1.2. What would be the quantity of food ordered by each individual, if A and B were presented with a single bill for the entire quantity of food consumed and each one of them paid for half of the amount billed?
3.1.3. Does each individual achieve a greater utility under the billing format in 3.1.1
or the billing format in 3.1.1?
3.1.4. Is there a way to split the single bill in 3.1.2 such that (i) individuals A and B respectively cover a fraction a and 1 -a of the bill, and that (ii) both individuals choose the same quantities of food as in 3.1.1?
