Consider Damon, an ancient Greek owner of a vineyard in Attica who employs a slave, Xanthias, to make wine. The master pays the slave a sum of A drachmas for subsistence.

Damon faces the following choices:

To incentivise Xanthias to work hard, he credibly promises that he will let him buy his freedom after working as a slave for the time it takes to make and sell wine and upon paying a price S.

If Xanthias works hard over the period of time and the weather is good, winemaking brings a revenue of (1+w)Y. The landlord takes Y and the slave takes the rest wY, which exceeds the price S. But if the slave works hard and the weather is bad winemaking brings a smaller revenue of Y(1r). Damon gives Xanthias A for subsistence, which is less than S, and keeps the rest of Y(1r). The cost of hard work for Xanthias is E.

If Damon does not offer Xanthias the prospect of freedom Xanthias will not work hard. If there is no prospect of freedom, Xanthias may either try to run away or resign to his state as slave and always accept a payment of A. If he tries to run away he incurs a cost of F and Damon will try to capture him an action, which costs Damon a sum of C. The probability that Xanthias escapes is G, so that the probability he is captured is (1G). If he is captured, his master does not feed him and sells him at the price S. If Xanthias escapes, he settles as a free person in a different place away from Attica, and upon practising his profession he enjoys a utility of (1+w)YE irrespective of the weather. If Xanthias stays put as a slave, winemaking brings to Damon a revenue of Z when the weather is good and Z(1k) when the weather is bad.

(a) Show the payoffs under the different outcomes described above

(b) If Xanthias is not offered the prospect of freedom, identify the condition which will lead him to try and escape

(c) Under what circumstances is Damon better off by offering Xanthias the opportunity to buy his freedom?

(d) Identify the maximum price S which will induce Xanthias to work hard on the hope of buying his freedom