Utility function?

Question

Carla wears disposable contact lenses. Each pair lasts her a month. The contact lens are sold in boxes, and each box has 3 pairs. She can’t wear the contacts without contact lens solution. Each bottle that she buys lasts her two months.

(i) What kind of preferences does Carla have for contacts and the solution? Write down the utility function representing these preferences. (Hint: write two bundles that Carla would consume. Plot these points on a graph of Contacts and Contact lens solution, and try to find a relationship for the line connecting these points. Use this relationship to specify the utility function)

i need help with it, finding what kind of preferences it represents and how to get the utility function

Answer 1

since the goods are complementary, it would have a kinked utility function, the type min[q1,q2]

Her utility can be thought of in terms of months' use.  Every two months she needs new solution, and every three months a new box.  Thus, in 6 months she will consume 3 bottles of solution and 2 boxes of lenses.  Since she will always consume in these ratios, her utility function is U=min{2q(lenses),3q(solution)}

Answer 2

you need a table that shows the relationship between boxes of lenses and bottles of solution.  Then you can graph the points and figure out what the shape and/or slope of the line means.    An easy way to do it is to multiply the units, so you get a pair that comes out even to start out with.

you get three months from a box of lenses and two months from a bottle, so to go 6 months, you need 2 boxes and three bottles.    at one month you need 1/3 of a box and 1/2 of a bottle, etc.  

Do this for each month of a year and then plot it on your graph and see what you get for a shape or a slop.  then look up the utility function on wiki and it will tell you how to read the line.