Question One ( 15 Marks) Consider two lotteries, $$ A $$ and $$ B $$. With lottery $$ A $$, there is a 0.90 chance that you receive a payofl of OFCFA and a 0.10 chance that $$ y $$ in receive a payoff of 40000 FCFA. With lottery $$ B $$. there is a 0.50 chance that you receive a pay ff of 3000 FCFA and a 0.50 chance that you receive a payoll of 5000 FCFA . $$ 3 $$ a) Verify that these two loiteries have the same expected value but that lottery $$ A $$ has a bigger variance than lottery $$ B $$. b) Suppose that your utility function is $$ \mathrm{U}=\sqrt{I+5 b 0} $$. Compute the expected utility of each lottery. Which lottery has the higher expected utility? Why? - c) Suppose that your utility function is $$ U=I+500 $$. Compute the expected utility of each lottery. If you have this utility function, are you risk averse, risk neutral, or risk loving? d) Suppose that your utility function is $$ U=(I+500)^{2} $$. Compute the expected utility of each lottery. If you have this utility function, are you risk averse, risk neutral. or risk loving? e) Suppose that your utility function is $$ U=\sqrt{ } $$. Compute the risk premium of lottery $$ \wedge $$ and B. Question Two ( 10 Marks) Suppose that you have a utility function given by the equation $$ U=\sqrt{501} $$. Consider a lottery that provides a payoff of OFCFA with probability 0.75 and $$ 20,000 \mathrm{FCFA} $$ with probability 0.25 . a) Sketch a graph of this utility function, letting I vary over the range 0 to 200. b) Verify that the expected value of this lottery is 5000 FCFA . c) What is the expected utility of this lottery? d) What is your utility if you receive a sure payoff of 5000 FCFA ? Is it bigger or smaller than your expected utility from the lottery? Based on your answers to these questions. are you risk averse? Question Two (5 Marks) Your current disposable income is 9.000000 FCFA. Suppose that there is a $$ 1 \% $$ chance that your house may burn down, and if it does, the cost of repairing it will be 8000000 FCFA, reduciny your disposable income to 1000000 FCFA . Suppose, too, that your utility function is $$ U=\sqrt{1} $$. a) Would you be willing to spend 50000 FCFA to purchase an insurance policy that

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**Question One (a)**

*Verify that these two lotteries have the same expected value but that lottery $$ A $$ has a bigger variance than lottery $$ B $$.*

Let's analyze both lotteries step by step.

### **Lottery $$ A $$:**

- **Outcomes and Probabilities:**
  - 0 FCFA with probability 0.90
  - 40,000 FCFA with probability 0.10

- **Expected Value ($$ E[A] $$):**
  $$
  E[A] = (0.90 \times 0) + (0.10 \times 40,000) = 0 + 4,000 = 4,000 \text{ FCFA}
  $$

- **Variance ($$ \text{Var}(A) $$):**
  
  First, calculate $$ E[A^2] $$:
  $$
  E[A^2] = (0.90 \times 0^2) + (0.10 \times 40,000^2) = 0 + 0.10 \times 1,600,000,000 = 160,000,000 \text{ FCFA}^2
  $$
  
  Then, calculate the variance:
  $$
  \text{Var}(A) = E[A^2] - (E[A])^2 = 160,000,000 - (4,000)^2 = 160,000,000 - 16,000,000 = 144,000,000 \text{ FCFA}^2
  $$

### **Lottery $$ B $$:**

- **Outcomes and Probabilities:**
  - 3,000 FCFA with probability 0.50
  - 5,000 FCFA with probability 0.50

- **Expected Value ($$ E[B] $$):**
  $$
  E[B] = (0.50 \times 3,000) + (0.50 \times 5,000) = 1,500 + 2,500 = 4,000 \text{ FCFA}
  $$

- **Variance ($$ \text{Var}(B) $$):**
  
  First, calculate $$ E[B^2] $$:
  $$
  E[B^2] = (0.50 \times 3,000^2) + (0.50 \times 5,000^2) = 0.50 \times 9,000,000 + 0.50 \times 25,000,000 = 4,500,000 + 12,500,000 = 17,000,000 \text{ FCFA}^2
  $$
  
  Then, calculate the variance:
  $$
  \text{Var}(B) = E[B^2] - (E[B])^2 = 17,000,000 - (4,000)^2 = 17,000,000 - 16,000,000 = 1,000,000 \text{ FCFA}^2
  $$

### **Comparison:**

- **Expected Values:**
  $$
  E[A] = 4,000 \text{ FCFA} \quad \text{and} \quad E[B] = 4,000 \text{ FCFA}
  $$
  Both lotteries have the **same expected value**.

- **Variances:**
  $$
  \text{Var}(A) = 144,000,000 \text{ FCFA}^2 \quad \text{and} \quad \text{Var}(B) = 1,000,000 \text{ FCFA}^2
  $$
  **Lottery $$ A $$ has a much larger variance** compared to Lottery $$ B $$, indicating higher risk.

### **Conclusion:**

- **Same Expected Value:** Both lotteries $$ A $$ and $$ B $$ have an expected value of 4,000 FCFA.
- **Different Variances:** Lottery $$ A $$ has a significantly higher variance (144,000,000 FCFA²) compared to Lottery $$ B $$ (1,000,000 FCFA²), meaning Lottery $$ A $$ is riskier.
**Question One (a)** *Both lotteries have the same expected value of 4,000 FCFA, but Lottery A is riskier with a higher variance of 144,000,000 FCFA² compared to Lottery B's variance of 1,000,000 FCFA².*

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