MSQE 2008 MICRO

[priyanka p]

on a tropical island there are 100 boat builders, numbered 1 trough 100. each builder can build upto 12 boats a year and each builder maximizes profit given the market price. let y denote the no. of boats build per year by a particular builder, and for each i , from 1 to 100, boat builder has a cost function Ci(y)= 11 + iy. asssume 11 to be quasi-fixed cost, that is , its paid only if firm produces a positive level of output. if the price of output is 25, how many builders would choose to produce a positive amt of output and how many boats would be built per year in total?

[Amit Goyal]

Boat builders will maximize profits:
ith boat builder has cost function Ci(y)= 11 + iy, for y > 0, Ci(0) = 0 its clear that boat builder i ≥ 25 will not build any boat because the price at which they will sell is 25 and MC exceeds 25. For i < 25, if a boat builder will produce positive amount he will produce upto his full capacity i.e. 12.
Profit of the ith builder must be ≥ 0 i.e. 25(12) - 11 - 12i ≥ 0 which implies i < 24.08, This inturn implies that 24 builders, i from 1 to 24, will produce a positive amount of output. Since each of them produces 12, the total no. of boats built per year is 288.