Read the following case "Fleet Sales Pricing at Fjord Motor" and answer parts 1, 2, and 3:

Fleet Sales Pricing at Fjord Motor The largest and most powerful automobile offered by the Fjord Motor Company in North America is the Coronet Elizabeth. It is sold through two different channels. Like Fjord's other models, it is sold to the public through Fjord's nationwide network of dealers. However, the Coronet Elizabeth is also a popular model for fleet purchases, particularly by corporations and police departments, and Fjord maintains a separate direct channel for fleet sales. Fjord defines fleet sales as those that include 10 or more vehicles. Most fleet sales are negotiated by Fjord's regional sales network. (A handful of very large deals involving hundreds of cars and mixed fleets are handled by a national sales team.) For the purposes of this case, you can assume that the only model in each fleet purchase is the Coronet Elizabeth, and that each bid proceeds with the buyers specifying the number of vehicles they wish to purchase, followed by Fjord submitting a sealed a bid for the order, specifying its price for satisfying the bid. Bids are competitive, but typically Fjord does not know who they are bidding against or the size of the bids submitted by the other competitors. After evaluating all bids, a buyer will tell Fjord whether Fjord has won the deal.' All fleet sales are for a plain vanilla (e.g., standard) version of the Coronet Elizabeth. This version costs Fjord $15,000 per vehicle to make, and the manufacturer's suggested retail price (MSRP) for it is $25,000. However, all fleet bids are somewhere between the cost (at which Fjord makes no margin) and the MSRP (at which the buyer receives no discount). Fjord makes about 4,000 fleet bids per quarter and wins almost 70% of them. According to Fjord management, about half the bids are to police departments, and half are to corporate fleet purchasing departments. Fjord management has recently become concerned that its fleet sales staff has been leaving money on the table through inconsistent and inaccurate bidding, and the company has therefore hired you as a consultant to help it institute a more rational pricing process for fleet sales. As part of this effort, Fjord has given you the data on the spreadsheet (see separate file). These data are the information available for all 4,000 bids made by the regional sales group during the last quarter of 2007, Unfortunately, due to difficulties reconciling the pricing database with the customer relationship management CRM database, these data do not include any information that would allow you to differentiate between bids to police departments and bids to corporations. 1. On the basis of the data in the spreadsheet, find a two-parameter logistic model that best estimates the probability of winning each bid as a function of the discount from list price, assuming a single price per unit will be offered for each bid. The model you are fitting is p(p) = 1/(1+exp(a +bp)), where p is the probability of winning the bid, and a and b are the parameters to be estimated. p is the price that Fjord bid on a deal expressed as a fraction of the MSRP of $25,000 per unit; that is, if Fjord bid $20,000 per unit, p would be equal to 20,000/25,000 or 0.8. What are the values of a and b that maximize the sum of log likelihoods? What is the optimum price Fjord should offer, assuming it is going to offer a single price for each bid? What would the expected total contribution have been for the 4,000 bids? How does this compare to the contribution that Fjord actually received? 2. Miraculously, Fjord discovers that bids 1 through 2,000 were to police departments, and that bids 2,001 through 4,000 were to corporate buyers. Taking advantage of this discovery, estimate two separate two-parameter logistic response functions, one for police departments and one for corporate buyers. The model you are fitting is the same as above, but the values of a and b for the police will be different than the values of a and b for corporate buyers. What are the corresponding values of a and b for each? What are the optimum prices Fjord should offer to the police? To corporate buyers? What would the expected contribution have been if Fjord had used the prices in the 4,000 bids in the database? What is the difference between the contribution actually received and the best that Fjord could do when it could not differentiate between the police and corporate buyers? (Hint: Excel Solver will work better if you use the values of a and b that you derived as answers in #1 as the starting values for solving the problems given in #2.) 3. As you continue your analysis, a senior sales manager tells you he believes that the size of the order is an important factor in determining price sensitivity. Specifically, he believes that customers who are placing larger orders are more sensitive to the price per unit. Add a single parameter c to your analysis to incorporate this potential effect. Your new model is p= 1/(1+exp(a+bp+cs)), where s is the number of vehicles in an order and c is the corresponding parameter that must be estimated. As before, calculate different values of a and b for police sales than for corporate sales, but calculate only a single value of c for both police and corporate sales. What is the resulting improvement in total log likelihood? How does this compare with the improvement from differentiating police and corporate sales? What are the optimal prices Fjord should charge for orders of 20 cars and for orders of 40 cars to police departments and to corporate purchasers, respectively? Fleet Sales Pricing at Fjord Motor The largest and most powerful automobile offered by the Fjord Motor Company in North America is the Coronet Elizabeth. It is sold through two different channels. Like Fjord's other models, it is sold to the public through Fjord's nationwide network of dealers. However, the Coronet Elizabeth is also a popular model for fleet purchases, particularly by corporations and police departments, and Fjord maintains a separate direct channel for fleet sales. Fjord defines fleet sales as those that include 10 or more vehicles. Most fleet sales are negotiated by Fjord's regional sales network. (A handful of very large deals involving hundreds of cars and mixed fleets are handled by a national sales team.) For the purposes of this case, you can assume that the only model in each fleet purchase is the Coronet Elizabeth, and that each bid proceeds with the buyers specifying the number of vehicles they wish to purchase, followed by Fjord submitting a sealed a bid for the order, specifying its price for satisfying the bid. Bids are competitive, but typically Fjord does not know who they are bidding against or the size of the bids submitted by the other competitors. After evaluating all bids, a buyer will tell Fjord whether Fjord has won the deal.' All fleet sales are for a plain vanilla (e.g., standard) version of the Coronet Elizabeth. This version costs Fjord $15,000 per vehicle to make, and the manufacturer's suggested retail price (MSRP) for it is $25,000. However, all fleet bids are somewhere between the cost (at which Fjord makes no margin) and the MSRP (at which the buyer receives no discount). Fjord makes about 4,000 fleet bids per quarter and wins almost 70% of them. According to Fjord management, about half the bids are to police departments, and half are to corporate fleet purchasing departments. Fjord management has recently become concerned that its fleet sales staff has been leaving money on the table through inconsistent and inaccurate bidding, and the company has therefore hired you as a consultant to help it institute a more rational pricing process for fleet sales. As part of this effort, Fjord has given you the data on the spreadsheet (see separate file). These data are the information available for all 4,000 bids made by the regional sales group during the last quarter of 2007, Unfortunately, due to difficulties reconciling the pricing database with the customer relationship management CRM database, these data do not include any information that would allow you to differentiate between bids to police departments and bids to corporations. 1. On the basis of the data in the spreadsheet, find a two-parameter logistic model that best estimates the probability of winning each bid as a function of the discount from list price, assuming a single price per unit will be offered for each bid. The model you are fitting is p(p) = 1/(1+exp(a +bp)), where p is the probability of winning the bid, and a and b are the parameters to be estimated. p is the price that Fjord bid on a deal expressed as a fraction of the MSRP of $25,000 per unit; that is, if Fjord bid $20,000 per unit, p would be equal to 20,000/25,000 or 0.8. What are the values of a and b that maximize the sum of log likelihoods? What is the optimum price Fjord should offer, assuming it is going to offer a single price for each bid? What would the expected total contribution have been for the 4,000 bids? How does this compare to the contribution that Fjord actually received? 2. Miraculously, Fjord discovers that bids 1 through 2,000 were to police departments, and that bids 2,001 through 4,000 were to corporate buyers. Taking advantage of this discovery, estimate two separate two-parameter logistic response functions, one for police departments and one for corporate buyers. The model you are fitting is the same as above, but the values of a and b for the police will be different than the values of a and b for corporate buyers. What are the corresponding values of a and b for each? What are the optimum prices Fjord should offer to the police? To corporate buyers? What would the expected contribution have been if Fjord had used the prices in the 4,000 bids in the database? What is the difference between the contribution actually received and the best that Fjord could do when it could not differentiate between the police and corporate buyers? (Hint: Excel Solver will work better if you use the values of a and b that you derived as answers in #1 as the starting values for solving the problems given in #2.) 3. As you continue your analysis, a senior sales manager tells you he believes that the size of the order is an important factor in determining price sensitivity. Specifically, he believes that customers who are placing larger orders are more sensitive to the price per unit. Add a single parameter c to your analysis to incorporate this potential effect. Your new model is p= 1/(1+exp(a+bp+cs)), where s is the number of vehicles in an order and c is the corresponding parameter that must be estimated. As before, calculate different values of a and b for police sales than for corporate sales, but calculate only a single value of c for both police and corporate sales. What is the resulting improvement in total log likelihood? How does this compare with the improvement from differentiating police and corporate sales? What are the optimal prices Fjord should charge for orders of 20 cars and for orders of 40 cars to police departments and to corporate purchasers, respectively