Capacity Choice with Expected Utility and Information Value
A contract manufacturer must choose capacity for a new product line: small, medium, or large. Profits in thousands of dollars depend on demand. For low demand (probability 0.3), medium demand (0.5), and high demand (0.2), the payoffs are: small 45, 48, 50; medium 10, 60, 120; large −60, 40, 200. The owner is risk-averse and uses the exponential utility function u(x)=1−exp(−x/40), where x is profit in thousands of dollars. A market-research firm offers a perfect-demand pilot study for $8,000 that would reveal the demand level before the capacity choice. The plant manager argues for the medium option because it has the highest expected monetary value and says the utility function is an unnecessary complication.
1. Compute the expected monetary value of each capacity option and the expected value of perfect information. What would a risk-neutral owner choose?
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EMV calculations (in $000):
Small: 0.3×45 + 0.5×48 + 0.2×50 = 13.5 + 24 + 10 = 47.5.
Medium: 0.3×10 + 0.5×60 + 0.2×120 = 3 + 30 + 24 = 57.0.
Large: 0.3×(−60) + 0.5×40 + 0.2×200 = −18 + 20 + 40 = 42.0.
A risk-neutral owner chooses medium with EMV $57,000.
With perfect information, choose the best action in each state: Low: max(45,10,−60)=45. Medium: max(48,60,40)=60. High: max(50,120,200)=200. Expected payoff with perfect information = 0.3×45 + 0.5×60 + 0.2×200 = 13.5 + 30 + 40 = 83.5.
EVPI = 83.5 − 57.0 = 26.5, or $26,500. The pilot costs $8,000, so its net value is at most $18,500 if it is truly perfect. The risk-neutral owner would buy it.
2. Compute the expected utility of each option and the certainty equivalent of the best option. Does risk aversion change the choice?
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Use u(x)=1−exp(−x/40). Key values: u(45)=0.6753, u(48)=0.6988, u(50)=0.7135; u(10)=0.2212, u(60)=0.7769, u(120)=0.9502; u(−60)=−3.4817, u(40)=0.6321, u(200)=0.9933.
Expected utilities: Small: 0.3×0.6753 + 0.5×0.6988 + 0.2×0.7135 = 0.2026 + 0.3494 + 0.1427 = 0.6947.
Medium: 0.3×0.2212 + 0.5×0.7769 + 0.2×0.9502 = 0.0664 + 0.3885 + 0.1900 = 0.6449.
Large: 0.3×(−3.4817) + 0.5×0.6321 + 0.2×0.9933 = −1.0445 + 0.3161 + 0.1987 = −0.5297.
Risk-averse ranking: small > medium > large. The small option wins despite having the lowest EMV among the serious contenders because medium has a bad low-demand payoff of 10 and large has a disastrous −60.
Certainty equivalent of small: solve 1−exp(−CE/40)=0.6947, so exp(−CE/40)=0.3053 and CE=−40 ln(0.3053)=$47,400. Medium's CE is −40 ln(1−0.6449)=$41,400. The owner would accept about $47,400 certain instead of the medium gamble. Risk aversion does change the decision.
3. Should the owner pay $8,000 for the perfect pilot? How does the answer depend on risk attitude and on whether the study is actually perfect?
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For a risk-neutral owner, the pilot is worth up to EVPI = $26,500, so $8,000 is easily justified. The expected net value is $26,500 − $8,000 = $18,500, ignoring the cost of delaying the decision.
For the risk-averse owner, the value of information is not simply the EMV EVPI; it should be computed in utility terms by comparing expected utility with and without the information, then converting the difference to a certainty equivalent. It can be higher than the risk-neutral EVPI because information reduces exposure to bad outcomes, but it need not be. The owner should compute that utility-based value before paying.
The study must also be scrutinized. 'Perfect' demand revelation is unrealistic; if the pilot has accuracy less than 100%, the owner needs posterior probabilities and the expected value of sample information, which is lower than EVPI. Also consider whether the pilot delays launch, whether demand levels are defined on the same time scale as the payoffs, and whether the $8,000 is sunk if the project is abandoned. Buy only if the expected utility gain exceeds the cost and the study design supports the claimed accuracy.