Answer Happy

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 q(p) = 1/(1+exp(a +bp)), where q 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 q =
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?
Extra credit: Calculate optimal prices for all order sizes from
10 through 60 vehicles for both police and corporate sales, and use
these prices to determine the total contribution margin Fjord would
have received if it had used these prices in the 4,000 historic
bids.