Problem:

Suppose Maria's preferences for goods A and B can be described by the following marginal utility schedules. Both product A and product B cost $2; Maria has allocated $8 to the purchase of these two products.

1. How should Maria allocate her $8 between A and B so as to achieve maximum utility?
2. Verify that the marginal utility per dollar of each good is the same at the utility maximizing bundle.
3. Suppose the price of A falls to $1. By comparing the marginal utility per dollar of her current purchases of A and B, should Maria purchase more of good A, more B, or does her current bundle still maximize her utility?
4. Assuming Maria still allocates $8 between the two goods, what amounts of goods A and B maximize her utility at these new prices?
5. From the exercises above, list two price/quantity combinations that lie on Maria's demand curve for good A.

Answer:

1. At the given prices, find the marginal utility per dollar for each good by dividing marginal utility by price. This results in the table below.

   	Product A	Product B
   Units of Product	MU/PA	MU/PA
   First	6	9
   Second	5	7.5
   Third	4	6
   Fourth	3	4.5
   Fifth	2	3
   Sixth	1	1.5

   
   The first unit of B yields the biggest increase in utility per dollar, and uses $2 from the budget. The second unit of B is still better than the first unit of A, and uses another $2. Either the first unit of A or the third unit of B yield the same increase in utility per dollar. With $4 left in the budget, purchase one of each. Three units of B and one unit of A maximize Maria's utility.
2. The marginal utility per dollar of A is 6 utils, the same as the third unit of good B.
3. At her current levels of consumption, the marginal utility per dollar of good A rises to $12, in excess of her marginal utility per dollar of good B, which remains at $6. She should purchase relatively more A.
4. Her current purchases now use only $7. She could use the extra dollar to purchase the second unit of good A, since its marginal utility per dollar (10) is now higher than the marginal utility per dollar of good B (still 6). Further, if Maria cuts back her purchases of B to two units, she sacrifices 12 utils of satisfaction but frees up $2. She could use this $2 to purchase the third and fourth units of A, gaining 8 utils of satisfaction on the third and 6 utils of satisfaction on the fourth, for a net gain of 2 utils of satisfaction on the switch. Her new consumption bundle is then 4 units of A and 2 units of B.
5. When the price of A is $2, Maria demands 1 unit; at a price of $1, Maria demands 4 units.