Problem:

Suppose Maria's preferences for eggs and milk can be described by the following marginal utility schedules. Both eggs and milk cost $4; Maria has allocated $16 to the purchase of these two products.

1. How should Maria allocate her $16 between eggs and milk 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 eggs falls to $2. By comparing the marginal utility per dollar of her current purchases of eggs and milk, should Maria purchase more eggs, more milk, or does her current bundle still maximize her utility?
4. Assuming Maria still allocates $16 between the two goods, what amounts of eggs and milk 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 eggs.

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.
   
   

   		Eggs		Milk
   Units of Product		MU/Peggs		MU/Pmilk
   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 milk yields the biggest increase in utility per dollar, and uses $4 from the budget. The second unit of milk is still better than the first unit of eggs, and uses another $4. Either the first unit of eggs or the third unit of milk yields the same increase in utility per dollar. With $8 left in the budget, purchase one of each to exhaust the budget. Three units of milk and one unit of eggs maximize Maria's utility.
2. The marginal utility per dollar on the first unit of eggs is 6 (= 24/$4) utils, the same as the marginal utility per dollar of the third unit of milk (=24/$4).
3. At her current levels of consumption, the marginal utility per dollar of eggs rises to $12, in excess of her marginal utility per dollar of milk, which remains at $6. She should purchase relatively more eggs.
4. Her current purchases now use only $14. She could use the extra two dollars to purchase the second unit of eggs, since its marginal utility per dollar (10) is now higher than the marginal utility per dollar of milk (still 6). Further, if Maria cuts back her purchases of milk to two units, she sacrifices 24 utils of satisfaction but frees up $4. She could use this $4 to purchase the third and fourth units of eggs, gaining 16 utils of satisfaction on the third and 12 utils of satisfaction on the fourth, for a net gain of 4 utils of satisfaction on the switch. Her new consumption bundle is then 4 units of eggs and 2 units of milk.
5. When the price of eggs is $4, Maria demands 1 unit; at a price of $2, Maria demands 4 units.