Consider the relationship between a principal and its agents, agents are divided into highly productive H type and low productive L type. When an agent performs work for x hours, he/she gets a reward w for it. However, the agents have no place to work other than this principal. H type agents produces 8√x results from x hours of work, while L type agents produces 4√x results. Corporate profit is the result of the agents’ result minus the remuneration w, and the utility of the agent is defined by u = w - x.
Problem;
The principal does not know the type of agents but knows that the ratio of H type is 1/3 and the ratio of L type is 2/3, find the combinations or contracts (x, w) that enable the screening of agents but at the same time, maximises the profit of the principal.