Let's solve each step in order for the given utility function \( U(I) = \log(I) \), income outcomes \( I_1 = 1000 \), \( I_2 = 4000 \), and \( p = 0.8 \): --- ### **1) Is this person risk-averse? If so, why?** #### **Explanation:** A person is **risk-averse** if their utility function is **concave** (second derivative is negative). For \( U(I) = \log(I) \): - First derivative: \( U'(I) = \frac{1}{I} \) - Second derivative: \( U''(I) = -\frac{1}{I^2} < 0 \) for all \( I > 0 \) **Conclusion:** Yes, the person is risk-averse because the utility function is strictly concave. --- ### **2) What is the expected income of this person?** #### **Calculation:** \[ E[I] = p \cdot I_2 + (1-p) \cdot I_1 = 0.8 \times 4000 + 0.2 \times 1000 = 3200 + 200 = 3400 \] **Final Answer:** \[ \boxed{3400} \] --- ### **3) What is the utility of expected income for this person?** \[ U(E[I]) = \log(3400) \] \[ \log(3400) \approx 8.126 \] **Final Answer:** \[ \boxed{8.126} \] --- ### **4) What is the expected utility of this person?** \[ E[U(I)] = p \cdot \log(I_2) + (1-p) \cdot \log(I_1) \] \[ = 0.8 \cdot \log(4000) + 0.2 \cdot \log(1000) \] \[ \log(4000) \approx 8.294 \] \[ \log(1000) = 6.908 \] \[ E[U(I)] = 0.8 \times 8.294 + 0.2 \times 6.908 = 6.635 + 1.382 = 8.017 \] **Final Answer:** \[ \boxed{8.017} \] --- ### **5) Compare 3) and 4). Which is larger? Why?** - Utility of expected income (3): **8.126** - Expected utility (4): **8.017** \[ 8.126 > 8.017 \] **Explanation:** For a risk-averse person, the utility of the expected income is **greater** than the expected utility, because the utility function is concave (Jensen's inequality). --- ### **6) What is the certainty equivalent income in this case?** Certainty equivalent (\( CE \)) is the guaranteed income that gives the same utility as the expected utility: \[ U(CE) = E[U(I)] \] \[ \log(CE) = 8.017 \] \[ CE = e^{8.017} \approx 3,034 \] **Final Answer:** \[ \boxed{3,034} \] --- ### **7) What is the risk premium in this case?** Risk premium (\( RP \)) = Expected income \(-\) Certainty equivalent \[ RP = E[I] - CE = 3,400 - 3,034 = 366 \] **Final Answer:** \[ \boxed{366} \] --- ### **8) Graphically show the risk premium with this utility function.** #### **Description of the graph:** - **X-axis**: Income (\( I \)), mark \( 1{,}000 \), \( 3{,}034 \), \( 3{,}400 \), \( 4{,}000 \) - **Y-axis**: Utility, mark \( U(1000) = 6.908 \), \( U(3034) = 8.017 \), \( U(3400) = 8.126 \), \( U(4000) = 8.294 \) - **Curve**: Plot \( U(I) = \log(I) \), a concave curve. - **Points**: - \( (1000, 6.908) \), \( (4000, 8.294) \): possible outcomes - \( (3400, 8.126) \): Expected income, utility of expected income - \( (3034, 8.017) \): Certainty equivalent income, matches expected utility - **Risk Premium**: The horizontal distance between \( 3,400 \) and \( 3,034 \) on the X-axis. **(You can draw this by hand or using software, but the above is how to label it.)** --- ### **9) Calculate the risk premium if p = 0.5.** **First, calculate expected income:** \[ E[I] = 0.5 \times 4000 + 0.5 \times 1000 = 2000 + 500 = 2500 \] **Expected utility:** \[ E[U(I)] = 0.5 \times \log(4000) + 0.5 \times \log(1000) \] \[ = 0.5 \times 8.294 + 0.5 \times 6.908 = 4.147 + 3.454 = 7.601 \] **Certainty equivalent:** \[ \log(CE) = 7.601 \implies CE = e^{7.601} \approx 2,002 \] **Risk premium:** \[ RP = 2,500 - 2,002 = 498 \] **Final Answer:** \[ \boxed{498} \] --- ### **10) Is risk premium in 9) smaller or larger than the one you get in 7)? Why? Briefly summarize the intuition behind this result.** - Risk premium with \( p = 0.8 \): **366** - Risk premium with \( p = 0.5 \): **498** \[ 498 > 366 \] **Explanation:** The risk premium is **larger** when \( p = 0.5 \) (greater risk), because the income is more uncertain. The more uncertain or risky the income, the more a risk-averse person is willing to pay to avoid risk. **Intuition:** Risk-averse individuals dislike risk. As the probability of the bad outcome increases (or probability of the good outcome decreases), risk increases, so the risk premium (amount willing to pay to avoid risk) rises. --- ## **Summary Table** | | \( p = 0.8 \) | \( p = 0.5 \) | |----------------|:-------------:|:-------------:| | \( E[I] \) | 3400 | 2500 | | \( E[U(I)] \) | 8.017 | 7.601 | | \( U(E[I]) \) | 8.126 | 7.824 | | Certainty Eq. | 3,034 | 2,002 | | Risk Premium | 366 | 498 | --- If you want a hand-drawn or software-generated graph, let me know!