A tugboat tows a sinking ship back to port after the tugboat captain and the ship captain sign a contract for the towing fee. When the ship gets back to port, the ship's captain refuses to pay, arguing that he signed the contract "under duress". The case goes to admiralty court.

If a ship leaves port on a sunny day, it will not get caught in a storm and become disabled. If it leaves port on a stormy day, the probability that it will get caught in a storm and become disabled is 0.002 (1 in 500). The probability that a tugboat is sufficiently close by that it can tow a distressed boat to port is 1 - 1/(T + 1), where T is the number of tugboats per ship. If there is no tugboat sufficiently close by to tow a distressed ship, the ship sinks at a cost of $5,000,000. A ship generates net revenue of $8500 per day when it leaves port (even if it gets distressed). Tugboats go out only on stormy days and do nothing other than roam the seas looking for distressed ships.. The cost per tugboat per stormy day is $2100.

The table below reports the net social benefit of a ship leaving port on a stormy day as a function of the integer number of tugboats per ship. An increase in the number of tugboats per ship decreases the probability that a ship will sink, and hence decreases the expected cost of a ship sinking. But the increase in the number of tugboats entails a higher resource cost.

no. of tugs/ship	0	1	2	3
net revenue/ship
expected cost of ship sinking
cost of tugboats/ship
net social benefit	-1500	1400	967	-300

a) Indicate how the number 967 for net social benefit in the above table is calculated.

Thus, on stormy days, the social optimum is to have ships leave port and for there to be one tugboat per ship.

In equilibrium the number of tugboats per ship is the maximum number such that tugboats make a profit.

The table below gives the equilibrium number of tugboats per ship and whether the ship will leave port, as a function of how much the ship captain pays the tugboat captain for the ship to be towed to shore. (M = millions of dollars).

payment	4.5M	4M	3.5M	3M	2.5M	2M	1.5M	1M	0.5M
exp cost of sunk ship
expected payment
no. of tugs	3	2	2	1	1	0	0	0	0
leave port	no	no	yes	yes	yes	no	no	no	no

b) Indicate how the indicated outcome with the payment of 3.5M by the ship's captain to the tugboat owner for towing the ship to shore is calculated.

It is assumed that the admiralty court chooses the damage award to incentivize the efficient allocation. When the damage award is too high, ship captains will choose not to leave port on stormy days since it is not profitable for them to do. When the damage award is too low, no tugs will choose to operate. This increases the probability of a ship sinking by enough that it is unprofitable for ships to leave port on stormy days.

c) What is the range of payments from the ship captain to the tugboat captain that the admiralty court should deem to constitute duress? Why?