Problem $1(10$ points$)$

Suppose an insurance company is negotiating the cost coverage of insulin with

a pharmaceutical company. The insurer's valuation for the insulin coverage

is represented by $U(q)=50ln(1+q),$ where $q$ indicates the number of insulin

doses covered. The pharmaceutical company incurs fixed costs of $$40,$ and

the production cost per insulin dose is $$0\mathrm{.}2$ for a less efficient production line

and $$0\mathrm{.}1$ for a more efficient one. The insurance company believes there is

a one$-$third likelihood that the pharmaceutical company will use the more

efficient production method.

$($a$)$ What are the first$-$best production levels? $(2$ points$)$

$($b$)$ What are the contracts to implement the first$-$best production levels?

$(2$ points$)$

$($c$)$ How much profit would the distributor make if the insurance company

offers a menu of contracts $(q_1^{\text{I}},t_1^{\text{I}}),(q_1^E,t_1^E)?(2$ points$)$

$($d$)$ What are the second$-$best production levels? $(2$ points$)$

$($e$)$ What is the menu of contracts for the second$-$best production levels?

$(1$ point$)$

$($f$)$ What is the information rent of an efficient agent for the menu of con$-$

tracts for the second$-$best production levels? Is this higher or lower than

the profit gained for the menu of contracts for the first$-$best production

levels? $(1$ point$)$