Nonlinear Programming in Price Optimization
March 31, 2012
By: Kristin Linstead (Evanto)
Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
2
Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
3
Dynamic pricing has become integral to the
practice in the optimization / OR space
•
“The better management of pricing is the fastest and most
cost-effective way to increase profits…a 1% improvement
in price creates an operating profit improvement of 11.1%”
– McKinsey & Company
•
“The CRM industry…[is] beginning to devote more
resources to price optimization...Gartner MarketScope
estimates the total worldwide revenue of the market for B2B
pricing software at $150 million as of 2007, and it’s growing
at 30% per year” – CRM Magazine
•
Industries highly susceptible to pricing pressure: cargo &
freight, entertainment & leisure, hospitality, media, and
passenger travel industries
–
American Airlines case study: the 1980’s price war against PeopleExpress
–
Revenue Management = special case of pricing problem with constrained supply
–
Price controls can be used dynamically to limit access to available seats
4
The list of potential objectives for a pricing
problem is endless…
•
Maximize total contribution
•
Survival
•
Maximize revenue
•
Avoid government investigation / intervention
•
Maximize long-run / long-run profit
•
•
Increase sales volume (quantity)
Obtain or maintain the loyalty and enthusiasm
of distributors and other sales personnel
•
Increase monetary sales
•
•
Increase market share
Enhance the image of the firm, brand or
product
•
Obtain a target rate of return on investment
•
•
Be perceived as “fair” by customers and
potential customers
Obtain a target rate of return on sales
•
•
Create interest and excitement about a product
Stabilize market or stabilize market price
•
•
Discourage competitors from cutting prices
Company growth
•
•
Use price to make the product “visible”
Maintain price leadership
•
•
Build store traffic
Desensitize customers to price
•
•
Discourage new entrants into the industry
Help prepare for the sale of the business
(harvesting)
•
Match competitors prices
•
Social, ethical, or ideological objectives
•
Encourage the exit of marginal firms from
the industry
•
Gain a competitive advantage
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Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
6
The Basic Price Optimization Problem
“In English”:
Total Contribution(1) = Unit Margin(2) × Demand for the product of a single
seller as a function of the price offered by that seller(3)
Definitions:
1) Total Contribution = m(p) = Sum of the margins of all products sold during a
time period
2) Unit Margin = (p – c) = Difference between the price at which a product is
sold and its incremental cost
3) Price-Response Function = d(p) = specifies how demand for a product varies
as a function of price
Mathematical Equation: m(p) = (p – c) × d(p)
7
The Price-Response Function…where
nonlinearity comes into play!
• There is one price-response curve associated with each combination of
product, market segment, and channel that relates price to demand
– Companies competing in the same market can have vastly different priceresponse curves
– In a perfectly competitive market, the price-response face by an individual
seller is a vertical line at market price (highly elastic demand)
8
The Price-Response Function (cont’d)
• Properties / Assumptions:
– Nonnegative
– Continuous
– Differentiable
– Downward sloping whenever d(p) > 0 (does not always hold)
• Common Price-Response Functions:
Linear: d(p) = D - mp
Constant-Elasticity: d(p) = Cp-e
Ce-(a+bp)
Logit: d(p) =
1+ e-(a+bp)
9
The Pricing Objective Function (Unconstrained)
Maximize: m(p) = (p – c) × d(p)
p* is the price
that maximizes
total
contribution
 This curve represents the maximum total contribution the supplier can
realize in the current time period
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We can now use standard optimization theory to
evaluate the objective function
1.
Take the derivative of our objective function with respect to price:
m¢(p) = d¢(p)(p - c)+ d(p)
2.
Set the derivative equation equal to zero
0 = d¢(p)(p - c)+ d(p)
3.
The price that solves the equation will maximize total contribution
d(p*) = -d¢(p*)(p*-c)
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Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
12
Example: Widget-making
A widget maker is looking to set the price of widgets for the current month.
Assume that the widget maker’s unit production cost c is a constant $1 per
widget and that his demand for the current month is governed by the linear
price-response function:
d(p) = (10 – 2p)
What is the objective function that will maximize total contribution?
Maximize -2p2 + 12p – 10
What is the equation that will be solved with optimal price, p*?
10 – 2p* = 2(p* – 1)
What is p*?
p* = 3
Note: All numbers in 00’s
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Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
14
We can derive three equivalent optimality
conditions using equation: d(p*) = -d¢(p*)(p*-c)
1.
Marginal Price = Marginal Cost
p* d¢(p*)+ d(p*) = cd¢(p*)
– Marginal Revenue (L.H.S. of equation) – additional revenue the seller could achieve
from a small increase in price
• Derivative of total revenue with respect to price
• Positive at low prices but negative at high prices
– Marginal Cost (R.H.S. of equation) – additional cost the seller would incur from a
small increase in price
• Always ≤ 0
– Total contribution is maximized at p* where marginal revenue = marginal cost
– Example:
 Key Take Away: If marginal revenue > marginal cost, the supplier can
increase contribution by increasing price (and vice versa)
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Optimality Conditions (cont’d)
2.
If the point elasticity at our current price is < 1, we can increase total
contribution by increasing price
m¢(p) = d(p) [1- e (p)] - d¢(p)c
– If d(p) > 0, then m’(p) will always be > 0 if ε(p) < 1
– Holds true until we reach a point where lost sales
outweigh increased unit margins
3.
Point
elasticity
Always ≥ 0 because
d’(p) ≤ 0
At optimum price (p*), the contribution margin ratio is equal to the
p*
reciprocal of elasticity
e (p*) =
( p *-c)
( p - c)
– The quantity
is known as the contribution margin ratio
p
1
– At p*, price elasticity = contribution margin ratio
 Any of the three optimality conditions can be used to compute the optimal
price; however, they may not hold if the price optimization is constrained
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Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
17
Pricing Case Study
• Background
− Company A is a global energy provider specializing in
uranium enrichment
− Marginal producer in market (high costs, low profit)
• Analysis
− Global estimation of demand levels of nuclear reactors
− Looked at historical prices and how price performs given supply/demand balance
• Findings
− Demand increasing, but finite supply – Company A controls the US market
− Company A set the market clearing price because they were the most expensive
− Win-win for all companies in the global market
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Outline
I.
“Real World” Application / Importance of Optimal Pricing
II.
The Basic Price Optimization Problem
III. Small-Scale Example of Price Optimization
IV. Optimality Conditions
V.
Pricing Case Study
VI. Appendix
19
Alternative Objective Functions
1.
Maximize Revenue
– Objective function: Maximize d(p) p
– Take derivative and set equal to zero: R¢( p)
ˆ = d¢( p)
ˆ pˆ + d( p)
ˆ =0
– pˆ = revenue-maximizing price
– This implies that pˆ solves: – We already learned that:
ˆ pˆ
d ¢( p)
ˆ =1
= e ( p)
ˆ
d( p)
Elasticity of priceresponse function
• Total contribution is maximized when marginal revenue = marginal cost…
• …and when marginal revenue = 0
• Therefore, the revenue-maximizing
pˆ
is lower than the contribution-maximizing
p*
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Alternative Objective Functions (cont’d):
Maximizing-Revenue Example
The CEO of widget-making company decides that the firm’s goal for the
next month will be to maximize revenue from widget sales as part of the
long-term strategy to increase market share.
d(p) = (10 – 2p)
What is the equation that will be solved with optimal price, pˆ ?
-4 pˆ +10 = 0
What is pˆ ?
pˆ = 2.50
Note: All numbers in 00’s
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Alternative Objective Functions (cont’d)
2.
Maximize weighted combination of revenue and contribution
– Most common approach is applying a weighting parameter: 0 < α < 1
– Objective function:
Z(p) = a (p - c)d(p)+ (1- a )pd(p)
– “Maximizing contribution with a discounted cost”
Weighted Combination
(Revenue &
Revenue-Maximizing
Contribution-Maximizing
≤
≤
Contribution)
Price ( pˆ )
Price ( p *)
Maximizing Price
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Sources
1.
Phillips, Robert. Pricing and Revenue Optimization. Stanford: Stanford
University Press, 2005. Print.
2.
Lagar, Marshall. "The Price is Right...You Hope." CRM Magazine. October
2008. Web. 22 March 2012.
<http://www.destinationcrm.com/Articles/Editorial/Magazine-Features/The-Pr
Is-Right...You-Hope-50751.asp&xgt;.
3.
“Pricing Objectives.” Wikipedia. Web. 23 March 2012.
<http://en.wikipedia.org/wiki/Pricing_objectives>.
4.
Taylor & Francis Group. Edited by A. Ravi Ravindran. Operations Research
Applications. Boca Raton: CRC Press, 2009. Print. [Originally printed in
Operations Research and Management Science Handbook (2008)].
5.
“Pricing and Revenue Optimization – Driving Value from CRM Investments.”
Pricing Society. Web. 23 March 2012.
<http://members.pricingsociety.com/articles/pricing-and-revenueoptimization-driving-value-from-crm-investments.pdf>.
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