C3016. Coursework 3
Deadline: To be handed by Thursday 16th November, 2006
Problem 1 (25% of the mark of this coursework)
For the third problem of the second coursework:
Write the optimal contract problem when effort is not verifiable. Prove which constraints are binding. Compute the optimal contract if effort is non-verifiable.
Problem 2 (25% of the mark of this coursework)
A manager can exert high (eh), medium (em) or low effort (el). His cost of effort will be v(eh)=2, v(em)=1, and v(el)=0. His remuneration is given by w. His reservation utility is zero. His utility function is U(w,e)=LOG10(w)-v(e) (remember that LOG10(0)= -INFINITY, LOG10(1)=0, LOG10(10)=1, LOG10(100)=2, etc…). Sales are given by X. The company shareholders’ utility function is X-w. The relation between X, effort and probabilities are given by:
Sales / effort / High effort / Medium effort / Low effortHigh / 3/4 / 1/2 / 0
Medium / 1/4 / 0 / 1/4
Low / 0 / 1/2 / 3/4
Which of the standard assumptions of the principal-agent model studied in the lectures fails in this problem?
Assuming that effort is verifiable, compute the optimal remuneration scheme that implements each level of effort
Assuming that effort is non-verifiable, compute the optimal remuneration scheme that implements each level of effort
Problem 3 (25% of the mark of this coursework)
A consumer’s initial wealth is £100. If an accident occurs, this consumer will lose £51 (l=51). If the accident does not occur, the consumer loses nothing (l=0). The probability of an accident depends on the level of effort exerted by the consumer. The consumer can exert either an effort of 0 (e=0) or an effort of 1 (e=1). There is only one insurance company from which the consumer can buy insurance. If the consumer does not accept the insurance contract offered by the insurance company, he has no other option but to remain uninsured. The insurance company is risk neutral and maximizes expected profits. The consumer’s utility function is w0.5-d(e) where w represents wealth net of any loss or insurance premium, and d(e) represents the cost of effort. The cost of effort is such that d(0)=0 and d(1)=1/3. Finally, suppose that the accident probabilities, conditional on effort, are given by the following entries:
No accident (l=0) / Accident (l=51)e=0 / 1/3 / 2/3
e=1 / 2/3 / 1/3
a) What is the consumer’s reservation utility?
b) Find the optimal contract, in terms of insurance premium and coverage, for the insurance company when effort is verifiable.
c) Show that the contract in part (b) will not induce high effort if effort is not verifiable.
d) Find the optimal insurance contract when effort is not verifiable.
e) Show that the symmetric information solution Pareto dominates that with asymmetric information.
Problem 4 (25% of the mark of this coursework)
Though cataracts surgery is carried out in dedicated operation theatres, cataracts surgery is only carried out in the mornings. Cataracts operation theatres are available for use in the afternoon. Visual capacity is measured on a scale from 0 (complete blindness) to 100 (full visual capacity). The National Health Service (NHS) recommends doctors to operate on only those patients with a visual capacity smaller than 50. However, the NHS cannot monitor compliance with this recommendation. Each operation takes 60+X minutes, where X is the difference between 100 and the patient’s visual capacity. The doctors that measure the patient’s visual capacity are the same ones that schedule the operations, and that carry them out. The NHS pays an annual salary of £50,000 to doctors in order to carry out cataracts operations in the mornings.
In an attempt to decrease waiting lists, the NHS will start a new initiative: pay £100 to each doctor for each operation carried out in the afternoon. Doctors decide what operations will be carried out in the morning and which in the afternoon.
a) Do you think that the pre-operation visual capacity of patients operated on in the morning is better or worse than that of those operated on in the afternoon? Justify your answer
b) Do you think that this system will certainly reduce the waiting lists for surgery cataracts?