Train the model with a subset of the data
Test the model on the remaining data (the validation set)
What data to choose for training vs. test?
In a time-series dimension, it is natural to hold out the last year (or time
period) of the data, to simulate predicting the future based on all past data.
In most settings, however, we’ll randomly select our training/test sets.
A class of methods to do many training/test splits and average over all the runs
Here is a simple example of 5-fold cross validation. Gives 5 test sets  5 estimates of MSE.
The 5-fold CV estimate is obtained by averaging these values.
Split the data up into K “folds”. Iteratively leave fold k out of the training data and use it to test.
The more folds, the smaller each testing set is (more training data), but the more times we need to
run the estimation procedure. Using rules of thumb like 5—10 folds is often utilized in practice. This
can be done with a simple for loop in R
For generalized linear models, the cv.glm() function can be used to perform k-fold cross
validation. For example, this code loops over 10 possible polynomial orders and computes the 10fold cross-validated error in each step
https://web.stanford.edu/~hastie/glmnet/glmnet_alpha.html
Features
Pricing Tier
Basic
Free
+ Premium 1
Good
+ Premium 2
Better
+ Premium 3
Best
Free: Fed EZ
Premium: State, Higher
income tax needs
Free: Read
Premium: Edit
Free: 10 articles
per month
Premium:
Unlimited
Free: 5GB
Premium:
Pay for storage
packages GB >5
Free: 12 Months
small VM
Premium: Pay-asyou-go IaaS
Free: 1 month
Premium: $10
per month
FB, Google, Yahoo
Free users
determine value.
Supplemental Biz
Model Required
LinkedIn, OKCupid, Evernote
Free users add
value to platform,
supplemental biz
models typically
necessary
Zynga, NYTimes, Skype, Adobe
TurboTax, WSJ
Product is free to
attract users,
ultimate goal is to
get most users to
pay
Most stuff
Required for high
marginal cost
products/services
Match.com, NetFlix,
Successful implementations
Failed or at risk
•
Free version “available forever”
•
Social networks
•
No premium versions avail
•
Search
•
Low-cost web publishing
•
Many smartphone apps
•
Network television
•
GroupOn
•
Travel booking
•
High-cost web publishing
•
User Access (Advertising)
•
User Data (Analytics)
•
User Control (Administration)
•
Any offering that did not achieve
sufficient scale
Conditions to Success
•
Very low marginal costs, fixed costs recovered with scale
•
Massive scale (typically 100s of Millions+)
•
High user retention rates
•
Strong network effects and/or Two-sided markets creating lock-in
Successful implementations
Failed or at risk
•
Power features
•
Dating sites
•
•
High capacity or usage
•
Social networks
•
•
Free version “available forever”
•
Communications
•
Adobe PDF
•
•
Job search
•
•
•
User Access (Advertising)
•
User Data (Analytics)
•
User Control (Administration)
Undifferentiated apps/software
Online version of traditional
newspapers
Zynga?
Spotify (unprofitable with 75M
users, 20M paid)?
Dropbox?
Conditions to Success
•
Similar to free conditions
•
Frictionless Path to Premium + Product Differentiation => Conversions
Successful implementations
•
•
•
•
Free version available is limited by time
(and/or capacity, features, or usage)
Most users will convert to premium once
they “understand” the value
Paths to conversion free to first tier and then
to more specialized offerings
Conversion is “within the product”
•
Free trials in many contexts
•
Cloud computing
•
Software (e.g. TurboTax, Office)
Failed or at risk
•
•
Undifferentiated products that
people abandon when the free
period expires
Premium version does not offer
enough value vs. the free alternatives
Conditions to Success
•
Learning and discovery (product-specific skills and comfort level)
•
Other product specific investments (e.g. engineering, stored files)
•
“Take a hostage”  create switching costs and dampen competition for
the high version(s) of the product
Successful implementations
•
•
•
Licenses and traditional goods: have lifetime
usage rights
•
Traditional economic output
Subscription/DRM limits to usage on time
dimension
Segmentation usually happens at time of
purchase
Failed or at risk
•
•
Goods with free substitutes (e.g.
newspapers vs. free websites, most
paid apps have very few downloads)
Stuff people don’t want
Conditions to Success
•
You make a good product (and/or have a great strategy)
?
Competitive/disruptive pressure of
free offerings in the space
Desire to take advantage of
benefits of freemium
Frictions to convert
1) entering payment information
2) paid features hard to find
3) higher tier versions cannot be
“unlocked” from within the product
• Un-targeted offers provide a freebie to “everyone”
-
Paying users have to support a larger base of freebies
• Offer restrictions, such as free trials that have eligibility triggers
based on platform usage can greatly limit costs, with minimal
damage to conversion rates
“Unlockable features” that can are surfaced within the “low version”
can be powerful drivers of conversions. Usage credits can be used to
allow free unlocking for limited time
• Traditional, profitable products generally had versions, e.g.
good-better-best, that are chosen at time of purchase.
Little/zero effort to upsell users from within the product
• “Grow up rich” because the “low version” is profitable
• In freemium, low version loses money! Have be very tactical about
within product upsell.
• Products that make freemium work efficiently eke out every last
conversion (targeted offers, unlocking features, frictionless
upgrades) as part of the core UX in the product
• Free & Hard Freemium success stories
• Requirements: massive scale, low marginal costs
• Good to have: strong networks effects, two-sided markets
• Supplemental monetization models required
• Know where you are and where you are going
• Legacy businesses are being pressured towards free, pushed “up the board”
• New businesses push for a larger fraction of paying user, trying to move “down the board”
• Maximize your current position
• Engineer the product to convert users to higher versions with identified segments in mind
• Minimize frictions along conversion paths
• “Freemium” is not a single strategy: different variations have fundamentally
different requirements for success
Introduce low
version
High price drops
Low Version
Attributes Improve
High Version
Attributes Improve



(maybe )
new







Impact to:
Acquisition Rate
Conversation Rate
Direct buy rate
Offsetting
Dynamics
Offsetting
Dynamics
Threshold Values by Varying Customer Lifetime
0.25
No free trial
0.2
0.15
Do free trial
0.1
0.05
0
0
10
20
30
40
Assumed Avg Customer Lifetime
50
60
Threshold Values Varying Marginal Cost
(lifetime=24 months)
0.25
Threshold
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
Marginal Costs
6
7
8
9
10
Jacob LaRiviere & Justin Rao
April 20, 2016
Econ 404, Spring 2016
Transfer Pricing
Interpretation versus Prediction
Monopolistic Competition
Monopolists, like all firms,
should price to maximize
profits.
As a result, the demand curve and costs
matter
Monopolist doesn’t have to worry about
competitors -> set Q such that MC = MR
max𝑞 𝜋 𝑞 = 𝑃 𝑞 𝑞 − 𝑇𝐶(𝑞)
Monopolist’s math
Maximizes profits (TR – TC) by setting a
quantity and charging needed price to have
market clear
e.g., price is a function on quantity: P(q)
and note the TR = P(q)*q
f.o.c.: 𝑃′ 𝑞 𝑞 + 𝑃 𝑞 − 𝑇𝐶 ′ 𝑞 = 0
𝑃′ 𝑞 𝑞 + 𝑃 𝑞 − 𝑀𝐶 𝑞 = 0
𝑃′ 𝑞 𝑞 + 𝑃 𝑞 = 𝑀𝐶 𝑞
𝑀𝑅(𝑞)
NOTE: P’(q) < 0 since demand slopes downward
𝑃′ 𝑞 𝑞
Intensive margin loss as q increases
𝑃 𝑞
Extensive margin gain as q increases
Upstream Monopolist’s math
Previously this is the point we introduced Lerner
equation. Now lets consider an upstream
monopolist selling to a downstream monopolist.
Can happen
- within a single firm (Amazon/MSFT)
- across firms (Content & Content Providers)
Upstream Monopolist’s math
Previously this is the point we introduced Lerner
equation. Now lets consider an upstream
monopolist selling to a downstream monopolist.
Can happen
- within a single firm (Amazon/MSFT)
- across firms (Content & Content Providers)
Rather than a competitive up and downstream
market lets just focus on the problem in market 1
- U1 is upstream firm
- D1 is downstream firm
Firm 1
Monopoly
Firm 2
Monopoly
D1’s problem: Same as before
D1 will take whatever the costs they pay to
U1 as given then price to maximize profits.
max𝑞 𝜋 𝑞 = 𝑃 𝑞 𝑞 − 𝑇𝐶(𝑞)
f.o.c.: 𝑃′ 𝑞 𝑞 + 𝑃 𝑞 − 𝑇𝐶 ′ 𝑞 = 0
𝑃′ 𝑞 𝑞 + 𝑃 𝑞 − 𝑀𝐶 𝑞 = 0
𝑃′ 𝑞 𝑞 + 𝑃 𝑞 = 𝐶𝑈1 =𝑃𝑈1
𝑀𝑅(𝑞)
NOTE: Irrespective of the upstream/downstream problem, this is bad for welfare [W(q)].
Social Optimum → W q = u q − c q → W ′ q
∗
= u′ q
∗
− c′ q
∗
= MB q
∗
− 𝑀𝐶 q
∗
=𝑃 𝑞
∗
−𝑐 =0
Monopoly equilibrium → 𝑃 𝑞𝑚 − 𝑐 = −𝑃′ 𝑞𝑚 𝑞𝑚
Extra term (
) represents the welfare gains to increasing output by an additional unit; creates deadweight loss.
U1’s problem: Novel
U1 understands the D1 will price as a monopolist.
As a result, their effective demand curve is the MR
curve of D1.
max𝑞𝐷1 𝜋 𝑞𝐷1 = 𝑃 𝑞𝐷1 𝑞𝐷1 − 𝑇𝐶 𝑞𝐷1
𝜋 𝑞𝐷1 = 𝑀𝑅 𝑞𝐷1 𝑞𝐷1 − 𝑇𝐶 𝑞𝐷1
∗
∗
∗
𝜋 𝑞𝐷1 = [𝑃′ 𝑞 (𝑐) 𝑞 +𝑃 𝑞 ]𝑞𝐷1 − 𝑇𝐶 𝑞𝐷1
𝑀𝑅(𝑞)
U1’s problem: Novel
U1 understands the D1 will price as a monopolist.
As a result, their effective demand curve is the MR
curve of D1.
max𝑞𝐷1 𝜋 𝑞𝐷1 = 𝑃 𝑞𝐷1 𝑞𝐷1 − 𝑇𝐶 𝑞𝐷1
𝜋 𝑞𝐷1 = 𝑀𝑅 𝑞𝐷1 𝑞𝐷1 − 𝑇𝐶 𝑞𝐷1
𝜋 𝑞𝐷1 = [𝑃′ 𝑞
∗
∗
∗
𝑐 𝑞 +𝑃 𝑞 ]𝑞𝐷1 − 𝑇𝐶 𝑞𝐷1
𝑀𝑅 𝑞
𝜋′
𝑞𝐷1 =
𝑀𝑅′
𝑞𝐷1 𝑞𝐷1 +
𝑀𝑅′
𝑞𝐷1 − 𝑀𝐶 𝑞𝐷1 = 0
MRDM
Retail
Price
Retail Demand
12
12
Quantity
Retail
Price
12
Marginal Revenue
12
Quantity
Retail
Price
12
4
Marginal Cost
QC = 8
12
Quantity
Retail
Price
12
Marginal Cost
QM = 4
QC = 8
12
Quantity
Retail
Price
Wholesale profits
12
Wholesale
Margin
8
Wholesale Price
4
Marginal Cost
QM = 4
QDM =2
QC = 8
12
Quantity
Retail
Price
Retail profits
12
Retail
Margin
10
8
Wholesale Price
4
Marginal Cost
QM = 4
QDM = 2
QC = 8
12
Quantity
Retail
Price
Surplus Under double marginalization
12
Wholesale Price
Marginal Cost
QDM
QM
QC
12
Quantity
Retail
Price
Surplus Under monopoly
12
Wholesale Price
Marginal Cost
QDM
QM
QC
12
Quantity
Retail
Price
Consumer Surplus Under monopoly
12
Wholesale Price
Fixed fee
Marginal Cost
QDM
QM
QC
12
Quantity
Durable good: Buy at t=1 or wait until t=2?
Durable good: Buy at t=1 or wait until t=2?
Non-Durable good: Buy once and hold or buy in each period?
x j
xi

n
n
0
 xi   j 1 ( p j  mc(i ))
 xi   j 1 ( p j  mc(i ))
pi
pi
p j