Question

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.

Answer 1

$$Q=\frac{1}{3}H+\frac{2}{3}L$$

$$w=\frac{8}{3}\sqrt{x}+\frac{8}{3}\sqrt{x}=\frac {16}{3} \sqrt {x}$$

$$u=w-x$$

$$u=\frac{16}{3}\sqrt{x}-x$$

$$\frac {\delta u}{\delta x}=\frac{8}{3\sqrt{x}}-1$$

$$x=\frac {64}{9}$$

$$w=\frac{16}{3}\times \sqrt{\frac{64}{9}}=\frac{128}{9}$$