Price Optimization Michel Bierlaire michel.bierlaire@epfl.ch - - PowerPoint PPT Presentation

Price Optimization

Michel Bierlaire

michel.bierlaire@epfl.ch

Transport and Mobility Laboratory

Price Optimization – p. 1/9

Introduction

• Choice model captures demand
• Demand is elastic to price
• Predicted demand varies with price, if it is a variable of the

model

• In principle, the probability to use/purchase an alternative

decreases if the price increases.

• The revenue per user increases if the price increases.
• Question: what is the optimal price to optimize revenue?

In short:

• Price↑⇒ profit/passenger↑ and number of passengers ↓
• Price↑⇒ profit/passenger↓ and number of passengers ↑
• What is the best trade-off?

Price Optimization – p. 2/9

Revenue calculation

Number of persons choosing alternative i in the population

ˆ N(i) =

S

• s=1

NsP(i|xs, pis)

where

• ps is the price of item i in segment s
• xs gathers all other variables corresponding to segment s
• the population is segmented into S homogeneous strata
• P(i|xs, pis) is the choice model
• Ns is the number of individuals in segment s

Price Optimization – p. 3/9

Revenue calculation

The total revenue from i is therefore:

Ri =

S

• s=1

NsP(i|xs, pis)pis

If the price is constant across segments, we have

Ri = pi

S

• s=1

NsP(i|xs, pi)

Price Optimization – p. 4/9

Price optimization

Optimizing the price of product i is solving the problem

max

pi pi S

• s=1

NsP(i|xs, pi)

Notes:

• It assumes that everything else is equal
• In practice, it is likely that the competition will also adjust the

prices

Price Optimization – p. 5/9

Illustrative example

A binary logit model with

V1 = βpp1 − 0.5 V2 = βpp2

so that

P(1|p) = eβpp1−0.5 eβpp1−0.5 + eβpp2

Two groups in the population:

• Group 1: βp = −2, Ns = 600
• Group 2: βp = −0.1, Ns = 400

Assume that p2 = 2.

Price Optimization – p. 6/9

Illustrative example

0.1 0.2 0.3 0.4 0.5 0.6 0.7 2 4 6 8 10 12 14 16 500 550 600 650 700 750 800 850 900 Share Revenues Price Share Revenues

Price Optimization – p. 7/9

Sensitivity analysis

• Parameters are estimated, we do not know the real value
• 95% confidence interval: [

βp − 1.96σ, βp + 1.96σ]

• Perform a sensitivity analysis for βp in group 2

Price Optimization – p. 8/9

Sensitivity analysis

200 400 600 800 1000 1200 1400 1600 1800 2 4 6 8 10 12 14 16 200 400 600 800 1000 1200 1400 1600 1800 Share Revenues Price beta=-0.04 beta=-0.07 beta=-0.1 beta=-0.13 beta=-0.16

Price Optimization – p. 9/9