In Problem Set #5, we studied the effect that premium loadings have upon the demand for insurance (see http://fin4335.garven.com/spring2018/ps5solutions.pdf for the solutions to this problem set). One of the important takeaways from Problem Set #5 was that other things equal, a more risk averse decision-maker will select a higher coinsurance rate for a given premium loading than a less risk averse decision-maker. Thus, if you have logarithmic utility; i.e., *U *= ln *W*, then your optimal coinsurance rate for a 20% premium loading is = 5/6, but if your utility is , then your optimal coinsurance rate for a 20% premium loading is = .9139.

The fact that is a more risk averse utility than *U *= ln *W* is apparent by comparing Arrow-Pratt risk aversion coefficients for these utilities (see pp. 16-26 of http://fin4335.garven.com/spring2018/lecture6.pdf for more on this topic). Specifically, the Arrow-Pratt risk aversion coefficient for *U *= ln *W* is equal to 1/*W*, and the Arrow-Pratt risk aversion coefficient for is equal to 2/*W. *Both of these utilities feature *decreasing absolute risk aversion*, which implies that if we were to increase initial wealth without making any further changes, then both of these utilities would imply lower coinsurance rates (although the decision-maker whose will still select a higher coinsurance rate then the decision-maker whose *U *= ln *W*).* *For example, suppose that we double initial wealth for both utilities without making any other changes; i.e., increase initial wealth from $1,000 to $2,000. Then, the optimal coinsurance rate for a 20% premium loading for *U *= ln *W* falls to = .614, and if , then the optimal coinsurance rate for a 20% premium loading falls to = .8007.

However, suppose that instead of only doubling initial wealth, we also double the state-contingent loss from $1,000 to $2,000, holding everything else constant. This will cause the coinsurance rates for both utilities to revert back to their original values of 5/6 and .9139 respectively. The reason why this occurs is because not only are these utilities characterized by decreasing absolute risk aversion, they also feature* constant relative risk aversion* (see the discussion on p. 25 of the http://fin4335.garven.com/spring2018/lecture6.pdf lecture note as to how relative risk aversion is calculated). Constant relative risk aversion implies that the *proportion* of wealth which the decision-maker is willing to put at risk does not change as wealth changes.