To: Mr. William Jaeger
From: Benjamin E. Hermalin, ph.d.
Re: Harvesting Decision
I have analyzed your harvesting decision problem. Figure 1 represents your problem in terms of a decision tree.
Botrytis
$67,200
[.4]
$24,000 - R
2 storm [.5]
4
wait
bottle
[.6] no Botrytis
1
[.5] sell as bulk
5
$12,000 + C
no storm
3
$42,000 good luck
[.4]
okay luck
[.4]
$36,000
bad luck
[.2]
$30,000
harvest now
$34,200
Figure 1: The harvesting decision represented as a decision tree.
In Figure 1, R denotes the monetary value of the loss of reputation you would suffer from bottling an inferior wine, while C denotes the cost saving from not bottling your own wine, should you choose to sell in bulk.
Assuming the information you gave me is correct (more on this later), R and
C are irrelevant to the problem. Given the information you provided, it seems safe to assume that C < $15, 000.∗ Even assuming the worst, R ≥ $12, 000 and
C = $0, the option of waiting has a greater expected value than the decision to harvest now: $35,640 versus $34,200.† If the worst is not true (i.e., R < $12, 000 or C > $0), the the option of waiting is even more valuable. Consequently, unless you strongly dislike risk, you should wait to harvest. Your expected gain from waiting is at least 4% (worst-case scenario). If the very best-case scenario in which the node (labeled as ①) is worth $28,200 (see Technical Appendix for underlying calculations), your expected gain would be at least 18%.
Because your winery is a partnership rather than a stock firm, I assume that risk is an issue for you. On the other hand, Riesling represents only 4% of your
∗ Otherwise, you not find it profitable to produce a $2.50 bottle of wine should the storm not occur and the acidity drops below 0.7%.
† I provide details in the attached technical appendix.
Copyright c 2004 Benjamin E. Hermalin. All rights reserved.
mba 201a Lecture 2 — Fall 2004
business, which suggests that this particular risk would not be very significant to you. Only a full evaluation of your attitudes toward risk—which I could conduct for an additional fee—would reveal whether this risk would be acceptable.
Finally, there is the accuracy of the numbers you gave me. I assume I can trust the prices you gave me. The analysis is largely independent of the storm probabilities. The analysis is, however, sensitive to the probability you gave me for the mold forming. If this probability were actually 34.8% or lower, then the expected-value maximizing decision would be to harvest now in the worse-case scenario. Technical Appendix
Basic Analysis
The decision tree shown in Figure 1 is read left to right. That is, it starts at the decision node labeled ⑤. The tree is solved backwards.
Starting at the decision node labeled ①, you will (should) bottle provided
$24, 000 − R > $12, 000 + C. The $24,000 is calculated as number of bottles,
12, 000 = 12 × 1000, times your estimate of a price of $2.00/bottle. Not being a vintner myself, I was unsure of how to account for the 5 to 10% swelling of the grapes, which lessened their concentration. If that means you get 5 to 10% more wine, then the $24,000 should be, at most, $26,400. The $12,000 figure comes from your estimate that grapes sold as bulk will yield only half as much revenue as grapes used in wine production. If the swelling effect also pertains to the grapes sold as bulk, then this number could be as large as $13,200. The value of node ① is thus at least $12,000 (if R ≥ $12, 000, C = $0, and the grape swelling yields no more wine). As noted, from the data provided, I feel I can assume the maximum possible cost savings from selling the wine as bulk is
$15,000. Hence, the maximum value of node ① is $28,200.
The value if the Botrytis forms is 70% of the 12,000 bottles (allowing for the
30% reduction in juice) times the $8/bottle price. That number is $67,200.
This makes the value of the chance node labeled ② equal to somewhere
