Kevin is a student who is yet to join a university in California. In the evening, he takes music lessons. Kevin has to decide on an apartment to rent

The apartment should be at a suitable distance from two places: his university, and from his music classes. The commuting time from apartments in four different locations to both destinations are shown in the table below. Given that Kevin's opportunity cost of time is $10 per hour, find his optimum choice using:
a) optimization in levels.
b) optimization in differences.

Apartment Commuting Time to University (hours per month) Commuting Time to Music Classes (hours per month) Rent ($ per month)
1 10 10 2,500
2 16 24 2,200
3 36 24 1,900
4 45 35 1,800

a) To optimize in levels, it is required to calculate the total cost of renting all apartments. This is shown in the table below.

Apartment Commuting Time to University (hours per month) Commuting Time to Music Classes (hours per month) Total Commuting Time (hours per month) Total Opportunity Cost of Commute per Month ($) Rent per Month ($) Total Cost per Month ($)
1 10 10 20 200 2,500 2,700
2 16 24 40 400 2,200 2,600
3 36 24 60 600 1,900 2,500
4 45 35 80 800 1,800 2,600

From the table above, it is seen that the total cost will be least for Kevin if he chooses to rent Apartment 3. Hence, optimizing in levels suggests that Apartment 3 is the optimum choice for Kevin.
b) To optimize in differences, it is required to calculate the marginal total cost of switching between each consecutive set of alternatives. This is shown in the table below.

Apartment Total Commuting Time (hours per month) Total Opportunity Cost of Commute per Month ($) Marginal Opportunity Cost of Commute per Month ($) Rent per Month ($) Marginal Rent per Month ($) Marginal Total Cost ($ per month)
1 20 200 - 2,500 0 -
2 40 400 200 2,200 -300 -100
3 60 600 200 1,900 -300 -100
4 80 800 200 1,800 -100 100

From the above table, Kevin is better off when he moves toward Apartment 3 and is worse off when he moves away from Apartment 3. Hence, according to the Principal of Optimization at the Margin it can be concluded that Apartment 3 is the optimum choice for Kevin.