Below are three microeconomics problems (or it could be insurance and investment problems as well) related to **absolute 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. Suppose the utility function is given by **u(W) = aW – bW ^{2}** (with a and b both positive). Does the function exhibit increasing or decreasing risk aversion?

**Answer:**

- First derivative of u(W) = aW – bW^{2}

**u’ = a – 2bW**

- Second derivative of u(W) = aW – bW^{2} or in other word first derivative of u’ = a – 2bW

**u” = -2b**

- Absoulute risk aversion -u”/u’

-u”/u’ = -2b/(a – 2bW)

So, when W increases, the denominator will decreases so that the coefficient of absolute risk aversion rises. Therefore the function exhibit **increasing risk aversion**.

2. If the rate of return on risky assets is a random variable R with mean R^>0 and variance σ^{2}_{R}, and if the individual’s wealth is W, what is the optimal amount of investment in risky assets?

**Answer:**

- let x be the amount invested in risky assets. The choice problem is

max E[a(W+xR) - b(W+xR)^{2}]

- The first order condition is:

E[aR-2bR(W+x*R)] = 0

aR^ – 2bWR^ – 2bx*(R^+σ^{2}_{R}) = 0

2bx*(R^+σ^{2}_{R}) = aR^ – 2bWR^

**x* = (aR^ – 2bWR^) / (2b(R^+σ ^{2}_{R}))**

x* is the optimal amount of investment in risky assets

3. Show that the optimal amount of risky investment is decreasing function of wealth.

- Optimal amount of investment in risky assets

x* = (aR^ – 2bWR^) / (2b(R^+ σ^{2}_{R}))

-First derivative of Optimal amount of investment in risky assets

**x*’ (W) = – 2bR^ / (2b(R^+ σ ^{2}_{R})) = -R^ / (R^+ σ^{2}_{R}) < 0**, so the optimal amount of risky investment is decreasing function of wealth

4. If the utility function is u(W) = -e^{-aW} so that the absolute risk aversion is constant. Show that the amount of investment in risky assets is independent of initial wealth**.**

**Answer:**

- let x be the amount invested in risky assets. The choice problem is

max E[-e^{-a(w+xR)}]

- First Order Condition:

E[aRe^{-a(w+xR)}] = 0

- Example

ae^{-aw} . E[Re^{-axR}] = 0

E[Re^{-axR}] = 0/ae^{-aw}

E[Re^{-axR}] = 0

The first order condition for x does not involve W. Therefore, the amount of investment in risky assets is not a function of initial wealth (independent of initial wealth**)**