Acquisition Pricing with Future Repricing
Opportunities
INFORMS Revenue Management Section Conference
June 5, 2006
Robert Phillips
[email protected]
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What is the basic problem?
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A seller is selling a service that is used on an on-going basis by a population
of subscribers.
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The seller offers the service initially for some period at an acquisition price.
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The seller has the opportunity to change the price at various times during the
duration of the subscription. Depending upon the industry, these repricing
opportunities may be:
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Periodic (e.g. subscription renewal)
Chosen pro-actively by the seller (e.g. to prevent attrition)
Triggered by buyer action (e.g. a late credit-card payment)
We concentrate on the periodic case.
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Applications
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Magazine Subscriptions
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B to C Services
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B to B Services
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Service Contracts of many types
Financial Services
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ISP
Satellite TV
Mobile Phone/Telecomm
Rent
Lines of Credit
Credit Cards
Loans
Certificates of Deposit (e.g. a 60-day CD)
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The most common case
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Existing customers are less price-sensitive than new prospects. That is for
any p > 0, ρe(p) > ρn(p), where:
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Possible Reasons:
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p = price
ρe(p) = probability that an existing customer will renew at price p
ρn(p) = probability than a new customer will purchase at price p
Selection: New customers uncertain about utility of service -- customers with low
utility learn quickly and drop out.
Switching Costs
Inertia
Risk-aversion: Known quality of service preferred to unknown alternative
Counter-example: Mortgage-refinancing.
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Previous Research
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Lewis (2005): Maximize customer value using a dynamic programming approach to
pricing magazine subscriptions dependent upon state where state is defined as:
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Months tenure for existing customers
Months lapsed for lapsed customers
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Formulated the problem as a dynamic program and solved using actual data.
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Assumed knowledge of state for both existing and lapsed customers and ability to
target prices accordingly.
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Other related work:
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Optimal catalog mailing (Bitran and Mondeschein [1995], Fusun and Shi [1998])
Customer Valuation (Gupta et. al. [2004])
Revenue Management with search and switching costs (van Ryzin and Liu [2005])
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Boiling the frog
The optimal policy for
magazine subscriptions is to
gradually increase renewal
price with tenure for both
current customers and lapsed
customers. Lewis (2005).
Figure from M. Lewis. (2005) Mgt. Sci. 51 6. pg 992.
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Property of Nomis Solutions Inc. - Confidential Material
Differentiators of our model
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In some sense simpler than Lewis:
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Does not assume differentiated pricing for lapsed customers
Reduced number of variables
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The desire is to gain some general insights and apply across various
applications
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We take an "equilibrium Markov process" approach rather than a dynamic
programming approach.
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This is a work in progress - we are beginning to apply to some actual
customer situations.
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The basic model
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N+1 states:
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State 0:
Not currently a customer
State i = 1, 2, ..., N-1: A customer with i periods of tenure
State N:
A customer with tenure > N periods*
N+1 prices:
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p0
pi, i = 1, 2, ..., N:
Price offered to non-customers (acquisition price).
Price offered to customer with i years of tenure.
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ρi(pi(t)) = Conversion rates in state i = 0, 1, 2, ... N-1 with ρ'i(pi) < 0 and 0 <
ρi(pi) < 1 for all i and p = (p0, p1, p2, ..., pN) > 0.
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πi(p) = Stationary probability of being in state i given price vector p.
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* Justified by the empirical observation that customer behavior tends to "converge" with longer tenure.
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A Markov Model
ρ1(p1)
ρ0(p0)
ρN-1(pN-1)
0
1-ρ0(p0)
1
2
N
1-ρ1(p1)
1-ρ2(p2)
1-ρN(pN)
πi(p) = Stationary probability of being in state i as a function of prices p.
N
TR(p) =
Σ
piρi(pi)πi(p)
i=0
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Property of Nomis Solutions Inc. - Confidential Material
ρN(pN)
Decreasing price-sensitivity with tenure
Linear price-sensitivities
Exponential price-sensitivities
1
1
aN
a2
a1
a0
Increasing
Tenure
Increasing
Tenure
...
0
P0
P1 P2
PN
ρi(pi) = ai(1-pi/Pi)
Page 10 Property of Nomis Solutions Inc. - Confidential Material
0
2
4
6
ρi(pi) = ai exp(-bipi)
8
10
Structure of Optimal Prices
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Acquisition price with re-pricing is always less than the optimal single price
without repricing.
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Not dependent on structure of price-response with tenure
Justifies "introductory discounts" and "promotions"
Optimal prices do not necessarily increase with tenure, even when
customers get more price-sensitive over time.
Page 11 Property of Nomis Solutions Inc. - Confidential Material
Three Policies to Compare
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Optimal Single Price: A single price is charged to all customers of all tenures.
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Optimal Independent Prices: The price is determined for each tenure of customers
that maximizes expected revenue from that tenure alone without considering the effect
on future pricing opportunities.
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Appropriate for companies who may not want to raise prices for loyal customers.
Possibly appropriate for organizations that separate acquisition from customer management
pricing.
Optimal Dynamic Price: The optimal price in each period considering the overall
effects on future pricing opportunities.
Page 12 Property of Nomis Solutions Inc. - Confidential Material
Policy Comparison
Comparison of policies for a realistic but contrived case: ρi linear with decreasing price
sensitivity for i = 0, 1, 2, ..., 5.
Revenue by Customer Tenure
0.8
Policy
Revenue
% of Opt.
Dynamic
1.515
100%
Single Price
1.495
98.7%
Optimal Ind.
1.404
92.7%
Revenue
0.6
Dynamic Optimum
0.4
Optimal Ind.
Single Price
0.2
0.0
0
1
2
3
4
5
Tenure
For this case, the optimum single price
significantly outperforms the static
optimization policy.
Page 13 Property of Nomis Solutions Inc. - Confidential Material
Independent pricing extracts too much
revenue at acquisition, reducing the ability
to gain revenue from longer-tenure
customers.
Policy Comparison (Continued)
Comparison of policies for a contrived case: ρi linear with decreasing price sensitivity
for i = 0, 1, 2, ..., 5.
Stationary Probabilities
Optimal Prices
14
0
Static
Independent
0
Price
10
Dynamic
log(Stationary Prob)
12
Dynamic
8
6
Single Price
4
1
2
3
4
5
-0.5
-1
-1.5
Single Price
-2
Dynamic
2
-2.5
0
0
1
2
3
4
5
Independent
Tenure (State)
Tenure (State)
Prices are typically lower under dynamic
optimization than static optimization.
Page 14 Property of Nomis Solutions Inc. - Confidential Material
Dynamic price-optimization results in more
demand in high-tenure states than static
price-optimization. In this case, singleprice policy can closely match the dynamic
policy.
How Many Prices are needed?
Example with exponential price response based on consumer lines of credit. Five
states (0,1,2,3,4). Price-sensitivity based on measurements from consumer loan
product.
Optimal Price tracks Under Alternative Policies
7
6
Price
5
Single Price
Two Prices
4
Three Prices
3
Four Prices
2
Five Prices
1
0
0
1
2
Tenure (State)
Page 15 Property of Nomis Solutions Inc. - Confidential Material
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Achievable revenue and number of price changes
There are decreasing marginal returns to additional price changes:
• Customers with longer tenure tend to "converge"
• There are fewer customers with longer tenure so the effect of price changes is smaller
Revenue as a percent of dynamic optimal
Revenue
100%
95%
90%
1
2
3
Number of Prices
Page 16 Property of Nomis Solutions Inc. - Confidential Material
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Alternative approach
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Model lifetime customer value "without replacement" - that is a customer who does not
renew is lost.
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Dynamic programming approach:
Vt = max ρt(pt)[pt + rVt+1]
pt
VT = max pTρT(pT)/(1 - rρT(pT))
pT
Vt = Value of having a customer of tenure t
r = Discount factor.
Hypothesis: Models with and without replacement should have similar solutions if
initial acquisition rate is quite small.
Page 17 Property of Nomis Solutions Inc. - Confidential Material
Discussion
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We have developed a relatively simple equilibrium model of optimal acquisition pricing
and re-pricing.
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Application of the model to realistic situations shows that the general (but not universal
policy) is to raise prices gradually with tenure.
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Much of the benefit comes from optimizing a single price considering the changing
price-sensitivity of customers with tenure.
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"Independent optimization" seems to be a particularly bad policy. It tends to over-price
all tenures relative to the optimal. This may have implications for the organization of
financial services into "acquisition" and "customer management" functions.
Page 18 Property of Nomis Solutions Inc. - Confidential Material
Issues and Research Opportunities
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The general trend is for prices to increase with tenure (although realistic cases can be
created in which price declines).
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Customers may have sensitivity to the "size" of increases that should be measured and
included.
The "boiling the frog" strategy goes against the general wisdom of rewarding loyal customers.
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One research issue is developing additional structural results -- what conditions on
price-response functions lead to prices rising with tenure. Can limits be calculated on
the size of potential benefits, etc?
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A second research issue is developing a more fundamental model of how customer
price response changes with tenure. Is there a model that incorporates selection
effects and switching costs in a consistent fashion that can be confirmed with real
world data.
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Compare model "with replacement" to models "without replacement" -- e.g. a
customer lost is lost forever.
Page 19 Property of Nomis Solutions Inc. - Confidential Material