Oct 222012
 

Below are two microeconomics problems (or it could be insurance and investment problems as well) related to relative risk aversion taken from the book The Structure of Economics: A Mathematical Analysis  3rd ed. by Eugene Silberberg and Wing Suen. These problems are include in the chapter Behavior Under Uncertainty. Absolute risk aversion has implications for the willingness of individuals to accept risk. The higher the coeffficient of absolute risk aversion, the higher the risk premium the individual is willing to pay. Relative risk aversion is absolute risk aversion times W, while W indicates initial Wealth. The higher the coeffficient of relative risk aversion, the higher the relative risk premium.

1. Verify that the function u(W) = W1-α/(1-α) has a constant coefficient or relative risk aversion equal to α

Answer:

- First derivative of u(W) =  W1-α/(1-α) is u’ = (1-α)W/(1-α) = W

- Second derivative of u(W) = W1-α/(1-α) or in other word First order condition (FOC) of W is u” = -αW-α-1

- Convex indifference curves imply risk aversion. A natural measure of the degree of risk aversion is the degree of convexity of indifference curve. The magnitude of second derivatives of indifference curve is proportional to -u”(W)/u'(W) or called coefficient of absolute risk aversion (Arroe-Pratt measure of absolute risk aversion). Relative risk aversion is -Wu”(W)/u'(W)

Thus, -Wu”/u’ = -W(-αW-α-1) / W = αW / W= α

2. Verify that the function u(W) = log W has a constant coefficient or relative risk aversion of 1

Answer:

- First derivative of u(W) =  log W is u’ = 1/W

- Second derivative of u(W) = log W or in other word First order condition (FOC) of 1/W is u” = -1/W2

- Convex indifference curves imply risk aversion. A natural measure of the degree of risk aversion is the degree of convexity of indifference curve. The magnitude of second derivatives of indifference curve is proportional to -u”(W)/u'(W) or called coefficient of absolute risk aversion (Arroe-Pratt measure of absolute risk aversion). Relative risk aversion is -Wu”(W)/u'(W)

Thus, -Wu”/u’= -W(-1/W2) / (1/W) = (1/W) / (1/W) = 1

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