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Problem 24

# Gordon Grimes, a self-employed consultant near Atlanta, received an invitation to visit a prospective client in Seattle. A few days later, he received an invitation to make a presentation to a prospective client in Denver. He decided to combine his visits, traveling from Atlanta to Seattle, Seattle to Denver, and Denver to Atlanta. Grimes received offers for his consulting services from both companies. Upon his return, he decided to accept the engagement in Denver. He is puzzled over how to allocate his travel costs between the two clients. He has collected the following data for regular round-trip fares with no stopovers: Grimes paid $$\ 900$$ for his three-leg flight (Atlanta-Seattle, Seattle- Denver, Denver-Atlanta). In addition, he paid $$\ 45$$ each way $$(\ 90$$ total ) for limousines from his home to Atlanta Airport and back when he returned. 1\. How should Grimes allocate the $$\ 900$$ airfare between the clients in Seattle and Denver using (a) the stand-alone cost-allocation method, (b) the incremental cost-allocation method, and (c) the Shapley value method? 2\. Which method would you recommend Grimes use and why? 3\. How should Grimes allocate the $$\ 90$$ limousine charges between the clients in Seattle and Denver?

Expert verified
Using the stand-alone cost-allocation method, Grimes should allocate the airfare between the clients based on the proportion of their round-trip fares ($$x$$ for Seattle and $$y$$ for Denver). For the incremental cost-allocation method, the airfare should be allocated $$300$$ for Seattle and $$600$$ for Denver. As for the Shapley value method, it should be an average of the stand-alone and incremental methods, resulting in $$\frac{1}{2} * \left(\displaystyle \frac{x}{x + y} * \900 + \300 \right)$$ for Seattle and $$\frac{1}{2} * \left(\displaystyle \frac{y}{x + y} * \900 + \600 \right)$$ for Denver. We recommend Grimes use the Shapley value method as it's a fair and balanced approach. For the limousine charges, we recommend allocating the charges equally at $$45$$ per client.
See the step by step solution

## Step 1: Calculate Round-Trip Distances

First, we need to find the round-trip distances between each pair of cities: - Atlanta (A) to Seattle (S) to Atlanta (A) - Atlanta (A) to Denver (D) to Atlanta (A) Since we are not given the actual distances, we will use the fares as a proxy for distance. - Round-trip fare from Atlanta to Seattle: $$\textrm{FA - S}$$ - Round-trip fare from Atlanta to Denver: $$\textrm{FA - D}$$

## Step 2: Determine Stand-Alone Method Allocation

For the stand-alone cost allocation method, we must allocate the $$900$$ fare based on the round-trip fares of each client as a proportion of the total fares. Let x and y be the round-trip fares from Atlanta to Seattle and Atlanta to Denver, respectively. Allocating Costs using Stand-Alone Method: - Seattle: $$\displaystyle \frac{x}{x + y} * \900$$ - Denver: $$\displaystyle \frac{y}{x + y} * \900$$

## Step 3: Determine Incremental Method Allocation

For the incremental cost allocation method, the journey to Seattle would be considered the first leg. Allocate the $$900$$ fare for the first leg, with the remainder split between the second leg from Seattle to Denver and the last leg from Denver to Atlanta. Allocating Costs using Incremental Method: - Seattle: $$\frac{1}{3} * \900 = \300$$ - Denver: $$\frac{2}{3} * \900 = \600$$

## Step 4: Determine Shapley Value Method Allocation

For the Shapley value method, we will average the costs allocated by the stand-alone and incremental methods. Allocating Costs using Shapley Value Method: - Seattle: $$\frac{1}{2} * \left(\displaystyle \frac{x}{x + y} * \900 + \300 \right)$$ - Denver: $$\frac{1}{2} * \left(\displaystyle \frac{y}{x + y} * \900 + \600 \right)$$

## Step 5: Recommendation of Allocation Method

Gordon Grimes should choose the Shapley Value method as it is a fair and balanced approach, considering the contributions of all clients in a proportional manner. This method takes into account the average of standalone and incremental methods, ensuring the fairest allocation to each client.

## Step 6: Allocating Limousine Charges

The limousine charges can be allocated using the same methods described above, or Grimes can allocate the limousine charges equally, as this cost is not dependent on the distance traveled to each client and would be incurred regardless of the client visited. We recommend allocating the limousine charges equally at $$45$$ per client as it works in the fairest manner for this situation.

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