Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will be equal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their income will be y − L. The chance of a safe return is 50%. Now suppose that there are two identical merchants, A and B, who are both risk averse expected utility maximisers with utility of income given by u(y) = ln y. The income of each merchant will be 8 if their own ship returns and 2 if it sinks. As previously, the probability of a safe return is 50% for each ship. However, with probability p ≤ 1/2 both ships will return safely. With the same probability p both will sink. Finally, with the remaining probability, only one ship will return safely. (iv) Compute the increase in the utility of each merchant that they could achieve from pooling their incomes (as a function of p). How does the benefit of pooling depend on the probability p? Explain intuitively why this is the case.
Consider a medieval Italian merchant who is a risk averse expected utility maximiser. Their wealth will be
equal to y if their ship returns safely from Asia loaded with the finest silk. If the ship sinks, their income
will be y − L. The chance of a safe return is 50%.
Now suppose that there are two identical merchants, A and B, who are both risk averse expected utility
maximisers with utility of income given by u(y) = ln y. The income of each merchant will be 8 if their
own ship returns and 2 if it sinks. As previously, the probability of a safe return is 50% for each ship.
However, with probability p ≤ 1/2 both ships will return safely. With the same probability p both will
sink. Finally, with the remaining probability, only one ship will return safely.
(iv) Compute the increase in the utility of each merchant that they could achieve from pooling their
incomes (as a function of p). How does the benefit of pooling depend on the probability p? Explain
intuitively why this is the case.
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