Question

Here is a question we could ask person n. For what magnitude of are you indifferent between having 5k for sure and having 1k with probability T1 and 10k with probability (1-m). Suppose person n answers π1-3. If we interpret that answer in terms of expected utility, then we have the equation where the second equality comes from the normalization, un (1k)-0 and un (10k)-1 Having determined that un(5k) - .7, we could ask other questions. Foir example, we could ask, for what magnitude of π1 is person n indifferent between having 3k for sure and having 1k with probability Ti and 5k with probability (1-T). Suppose they answer. π1-2. If we interpret that answer in terms of expected utility, then we have the equation where the second equality comes from the normalization, un(1k) = 0 and the previous result that un(5k) = .7. Notice that we are eliciting person n's attitude toward risk. There are no ri ght or wrong answers to these answers Exercise 2 Put yourself in the position of person n and pose and answer 4 questions that permit you to find your u function for 4 different magnitudes of r between 1k and 10k. Use some software to sketch those points of your u function.

Answer 1

Let x be the monetary value in $1000. Let u(x) indicate the utility of having $x (in thousands) (Here we are assuming that the monetary values are dollars although the question does not specify it)

We will first set the 2 values of the utilities at x=1 as u(x=1)=0 and u(x=10)=1. That is utility of $1k is set at 0 and utility of $10k is set at 1.

We want to fill in the following table, by eliciting responses for x=2,4,6, and 8 (in thousand dollars)

x ($1000)	u(x)
1	0
2
4
6
8
10	1

• Question 1: What value of $\pi_1$ that you are indifferent to receiving $4k for sure vs a lottery that will pay $1k with probability $\pi_1$ and $10k with probability $(1-\pi_1)$. A risk seeker will go for a higher value of $\pi_1$ (riskier bet) and a risk averse person will go for a lower value of $\pi_1$ (lower risk) (higher value is towards 1 and lower value is toward 0)
  • ans: $\pi_1=0.4$ (You can change these values, but you get what we are trying to achieve)

$u(x=4)=\pi_1u(1)+(1-\pi_1)u(10)=0.4\times 0+(1-0.4)\times1=0.6$

Hence the utility of $4k for me is 0.60

• Question 2: What value of $\pi_1$ that you are indifferent to receiving $2k for sure vs a lottery that will pay $1k with probability $\pi_1$ and $4k with probability $(1-\pi_1)$.
  • ans: $\pi_1=0.5$

$u(x=2)=\pi_1u(1)+(1-\pi_1)u(4)=0.5\times 0+(1-0.5)\times 0.6=0.3$

Hence the utility of $2k for me is 0.30

• Question 3: What value of $\pi_1$ that you are indifferent to receiving $6k for sure vs a lottery that will pay $4k with probability $\pi_1$ and $10k with probability $(1-\pi_1)$.
  • ans: $\pi_1=0.6$

$u(x=6)=\pi_1u(4)+(1-\pi_1)u(10)=0.6\times 0.6+(1-0.6)\times 1=0.76$

Hence the utility of $6k for me is 0.76

• Question 4: What value of $\pi_1$ that you are indifferent to receiving $8k for sure vs a lottery that will pay $6k with probability $\pi_1$ and $10k with probability $(1-\pi_1)$.
  • ans: $\pi_1=0.65$

$u(x=8)=\pi_1u(6)+(1-\pi_1)u(10)=0.65\times 0.76+(1-0.65)\times 1=0.844$

Hence the utility of $8k for me is 0.844

We get the following table of utilities

x ($1000)	u(x)
1	0
2	0.3
4	0.6
6	0.76
8	0.844
10	1

We plot this in excel using insert-->scatter--->lines

get the following raw graph

plot of u(x) 1.2 ︵ 0.8 0.6 0.4 0.2 0 4 10 x (thousands)

Since this utility is largely concave, this is a utility function for a risk averse person. But it turns up at higher payoffs, indicating a slightly higher risk apatite at higher payoffs.