An employer (P - principal) is looking for a new employee (A - agent) to work for him in his firm. Due to a Corona lockdown the employee will be working from home and the employer will therefore not directly observe how active (a) the employee is. Instead P will observe the total sales of his firm determined as q=a+ where e is a normally distributed random variable with mean 0 and variance o. Due to unions and boring legal stuff the employer is further only able to offer linear contracts of the form w =t+sq where t is thus a base amount and 8 governs how close the wage of the employee is linked to firm performance. The employer is risk neutral and interested in maximizing the expected profits of his firm. The agent has CARA preferences represented as: uw, a) = -e (2-4(a)) where w is the wage the agent gets when working and 7 >0 is the coefficient of absolute risk aversion. The function y(a) gives the cost of being active at level a and is given by v(a) = {ca? where e > 0. The outside option of the agent is given as: u(1) =-ee) (1) For a given (accepted) contract (t, x) formulate the maximization problem of the agent when deciding on the level of a. Hint: The agent is maximizing expected utility. When e is normally distributed with zero mean and variance on we have that for any (non-random), the following holds: E (CY) = (2) Solve for the level of a chosen by the agent, i.e. derive the first order condition and find an expression for a that satisfies the first order condition. (3) Having solved for a for a given contract (t, x), formulate the principal's problem under the second best. Hint: Instead of an IC constraint use the expression for a from above. (4) Argue that the PC constraint is binding under the second best, and use this to formulate the constraint as =t+ -- 1so? (5) Calculate the two first order conditions and interpret them. An employer (P - principal) is looking for a new employee (A - agent) to work for him in his firm. Due to a Corona lockdown the employee will be working from home and the employer will therefore not directly observe how active (a) the employee is. Instead P will observe the total sales of his firm determined as q=a+ where e is a normally distributed random variable with mean 0 and variance o. Due to unions and boring legal stuff the employer is further only able to offer linear contracts of the form w =t+sq where t is thus a base amount and 8 governs how close the wage of the employee is linked to firm performance. The employer is risk neutral and interested in maximizing the expected profits of his firm. The agent has CARA preferences represented as: uw, a) = -e (2-4(a)) where w is the wage the agent gets when working and 7 >0 is the coefficient of absolute risk aversion. The function y(a) gives the cost of being active at level a and is given by v(a) = {ca? where e > 0. The outside option of the agent is given as: u(1) =-ee) (1) For a given (accepted) contract (t, x) formulate the maximization problem of the agent when deciding on the level of a. Hint: The agent is maximizing expected utility. When e is normally distributed with zero mean and variance on we have that for any (non-random), the following holds: E (CY) = (2) Solve for the level of a chosen by the agent, i.e. derive the first order condition and find an expression for a that satisfies the first order condition. (3) Having solved for a for a given contract (t, x), formulate the principal's problem under the second best. Hint: Instead of an IC constraint use the expression for a from above. (4) Argue that the PC constraint is binding under the second best, and use this to formulate the constraint as =t+ -- 1so? (5) Calculate the two first order conditions and interpret them
