Exercise $4.$ Consider a market of used cars with different qualities. The qualities are

distributed on $0,1$ according to some cumulative distribution function $F,$ which we

assume is continuous and strictly increasing on $|$~$0,1$~$|.$ The owners know the quality of

their own cars, while the potential buyers do not know the quality. Assume owners'

reservation values of used cars are all $0.$ The value of a used car with quality $q$ for

a potential buyer is $q.$ Owners decide whether to accept a test at a cost cin$(0,1),$

by which the quality of his car will be fully revealed, and thus his car can be sold

at a price equals to its quality. In a market equilibrium, $($a$)$ there is one single price

for all cars without test, which equals to the average quality of these cars; and $($b$)$

a cutoff quality $q_c$ such that the owners of cars with quality strictly lower than $q_c$

will not accept the test, while the owners of cars with quality strictly larger than $q_c$

accept the test. Find the conditions on $F$ such that there exists at most one market

equilibrium.