Question

Suppose PDC is optimistic about the potential for the luxury high-rise condominium complex and that this optimism leads to an initial subjective probability assessment of 0.81 that demand will be strong (S₁) and a corresponding probability of 0.19 that demand will be weak (S₂). Assume the decision alternative to build the large condominium complex was found to be optimal using the expected value approach. Also, a sensitivity analysis was conducted for the payoffs associated with this decision alternative. It was found that the large complex remained optimal as long as the payoff for the strong demand was greater than or equal to $16.58 million and as long as the payoff for the weak demand was greater than or equal to -$17.32 million.

State of Nature Decision Alternative Strong Demand S1 Weak Demand S2 Small complex, di 9 8 Medium complex, d2 14 4 Large complex, d3 19 -7

1. Consider the medium complex decision. How much could the payoff under strong demand increase and still keep decision alternative d₃ the optimal solution? If required, round your answer to two decimal places.
   
   The payoff for the medium complex under strong demand remains less than or equal to $______million, the large complex remains the best decision.
   
2. Consider the small complex decision. How much could the payoff under strong demand increase and still keep decision alternative d₃ the optimal solution? If required, round your answer to two decimal places.
   
   The payoff for the small complex under strong demand remains less than or equal to $_______million, the large complex remains the best decision.

Answer 1

Solution:

Given

State of Nature
Decision Alternative	Strong Demand S1	Weak Demand S2
Small complex, d1	9	8
Medium complex, d2	14	4
Large complex, d3	19	-7

Expected Monetary value (EMV):

State of Nature
Decision Alternative	Strong Demand S1	Weak Demand S2	EMV
Small complex, d1	9	8	(0.81)(9)+(0.19)(8) = 8.81
Medium complex, d2	14	4

Decision alternative d₃ will remain the optimal decision alternative as long as EV(d₂) is less than or equal to the expected value of the optimal decision alternative,
EV(d₂) <= $14.06M
Let
S = the payoff of decision alternative d₂ when demand is strong
W = the payoff of decision alternative d₂ when demand is weak

Then

EV(d₂) = 0.81 x S + 0.19 x W

b) Assume the payoff under weak demand, W = $4M

EV(d₂) = 0.81 x S + 0.19 x ($4M) <= $14.06M
0.8 x S + $0.76M <= $14.06M
0.8 x S <= $14.06M - $0.76M

0.8 x S <= 13.30M
S <= 16.42M

The payoff for the medium complex under strong demand remains less than or equal to $16.42 million, the large complex remains the best decision.

For Small Complex Decision

Decision alternative d₃ will remain the optimal decision alternative as long as EV(d₁) is less than or equal to the expected value of the optimal decision alternative,
EV(d₁) <= $14.06M
Let
S = the payoff of decision alternative d₁ when demand is strong
W = the payoff of decision alternative d₁ when demand is weak

Then
EV(d₂) = 0.81 x S + 0.19 x W

Assume the payoff under weak demand, W = $8M

EV(d₂) = 0.81 x S + 0.19 x ($8M) <= $14.06M
0.8 x S + $1.52M <= $14.06M
0.8 x S <= $14.06M - $1.52M

0.8 x S <= 12.54M
S <= 15.48M

The payoff for the Small complex under strong demand remains less than or equal to $15.48 million, the large complex remains the best decision.

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