Michael Brown, president of Ivanhoe Always, agrees to construct a concrete cart path at Metlock Golf Club. Ivanhoe Always enters into a contract with Metlock to construct the path for $$200,000.$ In addition, as part of the contract, a performance bonus of $$34,000$ will be paid based on the timing of completion. The performance bonus will be paid fully if completed by the agreed$-$upon date. The performance bonus decreases by $$8,500$ per week for every week beyond the agreed$-$upon completion date. Michael has been involved in a number of contracts that had performance bonuses as part of the agreement in the past. As a result, he is fairly confident that he will receive a good portion of the performance bonus. Michael estimates, given the constraints of his schedule related to other jobs, that there is $55\%$ probability that he will complete the project on time, a $30\%$ probability that he will be $1$ week late, and a $15\%$ probability that he will be $2$ weeks late.

$($a$)$

Determine the transaction price that Ivanhoe Always should compute for this agreement.

Transaction price $

$($b$)$

Assume that Michael Brown has reviewed his work schedule and decided that it makes sense to complete this project on time. Assuming that he now believes that the probability for completing the project on time is $93\%$ and otherwise it will be finished $1$ week late, determine the transaction price. Use the expected value approach, as probabilities are available.

Transaction price $