The Black Knight

In a deep dark forest, there is a small bridge over a creek, which is the only way to get to the other side of the forest. Many knights want to cross this bridge, as legend has it that on the other side of the forest, the Holy Grail can be found. The Black Knight guards this bridge and is thus a monopolist over the only road through the forest. There are two types of knights: the rich Knights Templar ( = T ), and the poorer Knights of the Round Table ( = R). In order to let knights cross the bridge, the Black Knight can ask for two things: a payment of P golden florins, and for a sword fight of F minutes. Knights trying to cross the bridge dislike sword fighting, as it delays their path to the Grail. The Black Knight is risk neutral and only cares about florins, not about sword fighting. The knights who cross his path have the following utility functions

U(P, F) = G P F,

where R > T > 0. Both types of traveling knights have a reservation utility of zero. The proportion of knights of the round table is .

a. Draw the indifference curves of the Black Knight, the Knights Templar, and the Knights of the Round Table in P, F-space. Pay special attention to the shape, and think about how this influences the solution to the problem.

b. Before considering the usual individual rationality and incentive compatibility constraints, which other constraints should be added to this problem?

c. Assuming the Black Knight wants to let both knights pass, what are the optimal contracts?

d. In which case would the Black Knight only let one type of travelling knight pass? Which type? Explain.

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