IEOR E4601: Dynamic Pricing and Revenue Management
Dynamic Pricing and Revenue Management
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IEOR E4601: Dynamic Pricing and Revenue Management
Notes
• Late submissions will NOT be accepted.
• All questions in this homework must be individual work. You are allowed to discuss the
assignment with others but please mention in your write-up if you have discussed the solution
with someone.
• For the questions involving data, you should also submit your code as a pdf (Excel file,
MATLAB, Python or R functions) along with the answers.
Homework Exercises
1. (Maximum Likelihood Estimate for Logit Function of WTP) Consider the logit
demand model we discussed in class. Data {(p1, y1), . . . , (pN , yN )} are observed, where pi is
the price offered and yi is the indicator function of whether the product was sold at price pi.
Suppose the utility, u(p) for any customer at price p is given by a linear function β0 − β1p,
β0, β1 ≥ 0. For the logit demand model, the probability of purchase at price p, γ(p) is given
by
γ(p) =
exp(u(p))
1 + exp(u(p))
.
From the data {(p1, y1), . . . , (pN , yN )}, we can compute the MLE as follows
max
β0,β1
log
(
N∏
i=1
P(pi, yi)
)
= max
β0,β1
N∑
i=1
logP(pi, yi),
where P(pi, yi) is the probability of observing a sample (pi, yi) given parameters (β0, β1) and
is given by γ(pi)
yi(1− γ(pi))1−yi .
(a) Give the first order optimality conditions for β∗0 , β∗1 .
(b) Consider the data samples (price, sold/not sold) in InSample Q1.xlsx. Find the best
logit demand fit for the data assuming a linear utility function as discussed above.
(c) Compute the optimal price to maximize the expected revenue for the estimated model
from InSample Q1.xlsx. Use the data in OutOfSample Q1.xlsx to compute the revenue
for the optimal pricing.
2. (Empirical Willingness-to-pay distribution) We will use the data in InSample Q1.xlsx
and OutOfSample Q1.xlsx for this problem as well. However, instead of assuming a logit
demand model, we compute a non-parametric willingness-to-pay distribution using empirical
average. You should use the data in InSample Q1.xlsx to estimate the distributions and
compute the optimal price, and use the data in OutOfSample Q1.xlsx to compute the actual
revenue for the computed price. You can use any software including Excel, MATLAB, R or
Python to solve the estimation and optimization problems.
Suppose the willingness of paying this trip is a random variable W ∼ cdf F (·), here cdf refers
the cumulative density function of W .
1
(a) Use the data in InSample Q1.xlsx to estimate the cdf F (·). Note F is a stair-wise function
in this case. Draw a rough sketch to depict the function.
(b) Find the optimal price p∗ that maximizes the expected revenue. Suppose W1, . . . ,WN
are the willingness-to-pay data points in InSample Q1.xlsx. Then expected revenue for
any price p can be expressed as
1
N
·
N∑
i=1
p · 1(Wi ≥ p),
where 1(·) is an indicator function that evaluates to 1 if the condition is true and 0
otherwise.
(c) Suppose p∗ is the optimal price computed in the previous question. Compute the revenue
obtained by p∗ if the actual demand is given by data in OutOfSample Q1.xlsx.
(d) Now suppose there is a capacity constraint. You only have 10 spots to sell. Compute
the optimal price under the capacity constraint: N(1− F (p)) ≤ 10. Here N = 50 is the
total number of data points out of sample that can be interpreted as the market size.
Also, compute the revenue of the optimal price using data in OutOfSample Q1.xlsx.
3. (Pricing using a Linear demand function). For this question as well, we use the data
set in Demand Q3.xlsx. Suppose the demand function is linear: d(p) = a− bp, a, b > 0
(a) Use the data in Demand Q3.xlsx to estimate a, b using linear regression. You are given
50 historical (price, demand) pairs and you should divide the date into training and test
data and experiment with using regularization to get a linear model with good out of
sample performance.
(b) Find the optimal price p∗ that maximizes the expected revenue. We do not access to
out of sample data to compute the true revenue for the computed price. Therefore, it is
important to test whether the estimated linear function is a good predictor of demand
for test (out of sample) data points.