Lecture The expected value of guests is the average number of guests that would

Question

Lecture

The expected value of guests is the average number of guests that would arrive at the hotel if we repeat the situation many times. If we observed the hotel for 100 days, then 15% of the time we would see 98 people arrive. Since 15% of 100 days is 15 days, on 15 of the 100 days we would observe 98 guests arriving. On 25 days we would see 99 people, and so on. We could then add the total number of guests that arrive during those 100 days and then divide by the number of days to calculate the average number of guests per day which I will call X. This calculation would look like:

X = (98 * 15   +   99 * 25   + 100 * 30   +   101 * 20 +   102 * 10) / 100

We could calculate this to determine the numerical value. However, we can also distribute the 100 in the denominator and obtain

X = (98 * 15/100) + (99 * 25/100) + (100 * 30/100) + (101 * 20/100) + (102 * 10/100)

            = (98 * 0.15) + (99 * 0.25) + (100 * 0.30) + (101 * 0.20) + (102 * 0.10)

Or

X = (98 * 15%) + (99 * 25%) + (100 * 30%) + (101 * 20%) + (102 * 10%).

This is a simpler formula to apply generally; instead of needing to find a particular number of days such that each case occurs a whole number of days we can apply a formula similar to above. For each potential outcome, multiply the outcome by the probability of the outcome. Then add together those intermediate calculations to determine the average, or expected, value of the outcome.

Using Uncertainty in Calculations
In our hotel example the quantity of interest to the hotel is not the number of guests who arrive, but rather the profit. Profit can be calculated by examining the revenues and costs. Let us assume that the additional costs incurred from a single overbooking is $50. This number includes costs associated with finding the guest other accommodations, transportation costs to sent the guest to the other accommodations, an estimate of the loss of revenue from bad word-of-mouth advertising, and whatever other costs may be created by an overbooking. With this information we can determine expected revenues and expected costs. The expected revenue will be the sum of the room rate times the number of rooms rented times the probability of that many rooms being rented, or:

ER = $30 * 98 * 15% + $30 * 99 * 25% + $30 * 100 * 60% = $2983.50

Note that I combined the cases where 100, 101 and 102 guests arrive, since in all three of those cases exactly 100 rooms will be rented and the revenues will be identical. We could calculate the marginal costs associated with cleaning, guest services, and other such costs in a similar way. If the variable cost per occupied room is $5, then the expected variable costs will be $497.25. The expected losses from overbooking can be estimated as

EC = $0 * 70% + $50 * 20% + $100 * 10% = $20.

Here there is a 70% chance the hotel is not overbooked, or in other words only 98, 99 or 100 guests arrive and thus there is room for all. A 20% chance exists that the hotel will be overbooked by one room and a 10% chance two people will not find lodgings available.

The profit maximizing price is calculated by determining the marginal revenue and marginal cost associated with a small change in the price and finding the price at which the marginal cost is equal to the marginal revenue. When uncertainty is present, this rule is changed to finding the expected values of the marginal cost and marginal price. If the price is adjusted slightly, the probability that a certain number of guests arrive is likely to change. Therefore, if the price of the room is changed, the calculations outlined above can be recomputed with the new outcomes and probabilities and the change in expected revenue and expected costs can be determined.

For example, suppose lowering the price to $29.90 results in 99 guests arriving 20% of the time, 100 guests 35% of the time, 101 guests 20% of the time, 102 guests 15% of the time, and 103 guests 10% of the time. In that case, the expected revenue would be

ER = $29.90 * 99 * 20% + $29.90 * 100 * 80% = $2984.02.

So the expected marginal revenue is positive since the expected revenue in this case is larger. The variable costs per occupied room would be $499. Immediately we can see that the expected marginal cost ($499 - $497.25 = $1.75) is larger than the expected marginal revenue ($0.52). The change in expected costs from overbooking simply adds more to the marginal cost of decreasing the room rate. To be complete, the expected costs from overbooking are now:

EC = $0 * 55% + $50 * 20% + $100 * 15% + $150 * 10% = $40.

Thus the motel should not lower its price to increase revenue. This analysis suggests that the room rate should be decreased in order to increase profit but the probability of particular numbers of guests arriving would need to be known before definitively making that statement.

Scenario 3 (length: as needed)
Suppose the hotel in the lecture example raised its price from $30 to $30.50. With the new price, the hotel expects 96 guests to arrive 5% of the time, 97 guests 10% of the time, 98 guests 20% of the time, 99 guests 30% of the time, 100 guests 25% of the time and 101 guests 10% of the time. The variable costs per occupied room and overbooking costs are the same as in the lecture.

Explanation / Answer

Expected Number of Room Rented: (96)(.05) + (97)(.1) + (98)(.2) + (99)(.3) + (100)(.25) + (100)(.1) = 98.8

Expected Revenue = (98.8)($30.50) = $3013.40

Expected Variable Costs = (98.8)($5) = $494

Expected Overbooking = (1)(.1)($100) = $10

Using marginal analysis, should the hotel raise its price? Explain your answer.

Suppose a change in price to $31 results in 96 guest arriving 10% of the time, 97 guest 15% of the time, 98 guests 20% of the time, 99 Guest 30% of the time, 100 guests, 15% of the time and 101 guests 10% of the time. The the Expected rates become:

Expected Number of Room Rented: (96)(.10) + (97)(.15) + (98)(.2) + (99)(.3) + (100)(.15) + (100)(.1) = 98.45

Expected Revenue = (98.45)($31) = $3051.95

Expected Variable Costs = (98.45)($5) = $492.45

Expected Overbooking = (1)(.1)($100) = $10

In this case, Marginal Revenue increased and Marginal costs decreased. This results in a higher Marginal profit. As a result, it makes sense for the hotel to raise the price to $31

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