Let's solve each part step-by-step: --- ## **Given Data** ### **Payoff Table** | Product | Favorable (0.2) | Stable (0.7) | Unfavorable (0.1) | |----------|-----------------|--------------|-------------------| | Widget | $120,000 | $70,000 | -$30,000 | | Hummer | $60,000 | $40,000 | $20,000 | | Nimnot | $35,000 | $30,000 | $30,000 | --- ## **(a) Compute the expected value for each decision and select the best one.** **Expected Value (EV) formula:** \[ EV = P_1 \times V_1 + P_2 \times V_2 + P_3 \times V_3 \] where \( P \) = probability, \( V \) = value ### **Widget:** \[ EV_{Widget} = (0.2 \times 120,000) + (0.7 \times 70,000) + (0.1 \times -30,000) \] \[ = 24,000 + 49,000 - 3,000 = \boxed{70,000} \] ### **Hummer:** \[ EV_{Hummer} = (0.2 \times 60,000) + (0.7 \times 40,000) + (0.1 \times 20,000) \] \[ = 12,000 + 28,000 + 2,000 = \boxed{42,000} \] ### **Nimnot:** \[ EV_{Nimnot} = (0.2 \times 35,000) + (0.7 \times 30,000) + (0.1 \times 30,000) \] \[ = 7,000 + 21,000 + 3,000 = \boxed{31,000} \] ### **Best Decision:** **Widget** has the highest expected value (**$70,000**). --- ## **(b) Develop the opportunity loss table and compute the expected opportunity loss for each product.** ### **Step 1: Find the best payoff for each state of nature:** - Favorable: max(120,000; 60,000; 35,000) = **120,000** - Stable: max(70,000; 40,000; 30,000) = **70,000** - Unfavorable: max(-30,000; 20,000; 30,000) = **30,000** ### **Step 2: Construct Opportunity Loss Table:** \[ \text{Opportunity Loss} = \text{Best Payoff for State} - \text{Payoff for Product} \] | Product | Favorable | Stable | Unfavorable | |----------|-----------|--------|-------------| | Widget | 0 | 0 | 60,000 | | Hummer | 60,000 | 30,000 | 10,000 | | Nimnot | 85,000 | 40,000 | 0 | ### **Step 3: Compute Expected Opportunity Loss (EOL) for Each Product** #### **Widget:** \[ EOL_{Widget} = (0.2 \times 0) + (0.7 \times 0) + (0.1 \times 60,000) = 0 + 0 + 6,000 = \boxed{6,000} \] #### **Hummer:** \[ EOL_{Hummer} = (0.2 \times 60,000) + (0.7 \times 30,000) + (0.1 \times 10,000) \] \[ = 12,000 + 21,000 + 1,000 = \boxed{34,000} \] #### **Nimnot:** \[ EOL_{Nimnot} = (0.2 \times 85,000) + (0.7 \times 40,000) + (0.1 \times 0) \] \[ = 17,000 + 28,000 + 0 = \boxed{45,000} \] ### **Best Decision:** **Widget** has the lowest expected opportunity loss (**$6,000**). --- ## **(c) Determine how much the firm would be willing to pay for perfect information (Expected Value of Perfect Information, EVPI).** \[ \text{EVPI} = \text{Expected Value with Perfect Information (EVwPI)} - \text{Maximum Expected Value (EV)} \] ### **Step 1: EVwPI** For each state, choose the best payoff and multiply by the probability: \[ EVwPI = (0.2 \times 120,000) + (0.7 \times 70,000) + (0.1 \times 30,000) \] \[ = 24,000 + 49,000 + 3,000 = \boxed{76,000} \] ### **Step 2: Maximum Expected Value (from part a):** \[ = 70,000 \quad (\text{Widget}) \] ### **Step 3: Calculate EVPI** \[ \text{EVPI} = 76,000 - 70,000 = \boxed{6,000} \] --- ## **Summary of Answers** **a.** Expected values: - Widget: **$70,000** (Best Decision) - Hummer: $42,000 - Nimnot: $31,000 **b.** Expected Opportunity Losses: - Widget: **$6,000** (Best Decision) - Hummer: $34,000 - Nimnot: $45,000 **c.** The firm would be willing to pay up to **$6,000** for perfect information about future market conditions. --- **If you need the tables typed up or any clarifications, let me know!**