Effect

of

electronic

secondary

markets

on

the

supply

chain

IT-based

price

discrimination

and

customization,

estimating

demand

and

efficiency

in

electronic

markets

and

in

the

economics

of

IT

security.

His

recent

work

has

estimated

product

cannibalization

and

social

welfare

in

industries

transformed

by

electronic

used

good

markets,

the

profitability

of

personalized

pricing

and

quality

customization,

and

the

economic

incentives

for

security

information

sharing.

His

current

work

examines

pricing

and

product

proliferation

in

software

markets,

and

dimensions

of

reputation

in

e-secondary

markets.

His

research

has

appeared

or

is

forthcoming

in

Management

Science,

Information

Systems

Research,

and

Journal

of

Management

Information

Systems.

His

research

interests

include

consumers'

use

of

new

technologies

such

as

search

engines

and

peer-to-peer

networks,

and

empirical

and

analytical

models

of

software

security

and

vulnerabilities.

His

current

work

on

e-business

examines

loyalty

on

the

Internet,

competition,

and

effect

of

used

good

markets

on

retailers

and

suppliers.

His

recent

papers

have

studied

the

impact

of

patching

on

software

quality,

markets

for

software

vulnerabilities,

and

so

on.

His

research

has

appeared

in

Management

Science,

Journal

of

Marketing

Research,

and

Journal

of

Management

Information

Systems.

RAMAYYA

KRISHNAN

is

the

W.W.

Cooper

and

Ruth

F.

Cooper

Professor

of

Information

Systems

at

the

Heinz

School

of

Public

Policy

and

Management

at

Carnegie

Mellon

University.

His

research

interests

span

technology,

policy,

and

business

issues

related

to

Internet-enabled

systems.

He

is

presently

codepartment

editor

for

Information

Systems

at

Management

Science.ABSTRACT:

We

present

a

model

to

investigate

the

competitive

implications

of

electronic

secondary

markets

that

promote

concurrent

selling

of

new

and

used

goods

on

a

supply

chain.

In

secondary

markets

where

suppliers

cannot

directly

utilize

used

goods

for

practicing

intertemporal

price

discrimination

and

where

transaction

costs

of

resales

are

negligible,

the

threat

of

cannibalization

of

new

goods

by

used

goods

becomes

significant.

We

examine

conditions

under

which

it

is

optimal

for

suppliers

to

operate

in

such

markets,

explaining

why

these

markets

may

not

always

be

detrimental

for

them.

Intuitively,

secondary

markets

provide

an

active

outlet

for

some

highvaluation

consumers

to

sell

their

used

goods.

The

potential

for

such

resales

leads

GHOSE,

TELANG,

AND

KRISHNAN

to

an

increase

in

consumers'

valuation

for

a

new

good,

leading

them

to

buy

an

additional

new

good.

Given

sufficient

heterogeneity

in

consumers'

affinity

across

multiple

suppliers'

products,

the

"market

expansion

effect"

accruing

from

consumers'

cross-product

purchase

affinity

can

mitigate

the

losses

incurred

by

suppliers

from

the

direct

"cannibalization

effect."

We

also

highlight

the

strategic

role

that

the

used

goods

commission

set

by

the

retailer

plays

in

determining

profits

for

suppliers.

We

conclude

the

paper

by

empirically

testing

some

implications

of

our

model

using

a

unique

data

set

from

the

online

book

industry,

which

has

a

flourishing

secondary

market.KEY

WORDS

AND

PHRASES:

electronic

markets,

information

goods,

market

segmentation,

quality

degradation,

supply

chain,

used

goods

markets.INFORMATION

TECHNOLOGY

(IT)

REDUCES

the

search

and

transaction

costs

for

buyers

and

sellers

to

locate

and

trade

products,

and

can

thereby

facilitate

the

creation

of

technology-mediated

electronic

exchanges.

These

Internet-based

exchanges

allow

sellers

to

easily

reach

a

worldwide

market

and

allow

buyers

to

easily

locate

items

that

frequently

would

be

unavailable

in

traditional

physical

stores.

Consumer-to-consumer

exchanges

represent

one

prominent

area

where

the

low

search

and

transactions

costs

in

IT-enabled

markets

have

enabled

product

exchanges

that

would

not

have

been

viable

in

a

comparable

bricks-and-mortar

environment.

The

recent

establishment

of

electronic

exchanges

by

some

retailers

to

sell

used

goods

has

resulted

in

a

contentious

debate

about

their

potential

affect

on

suppliers.

For

instance,

publishers

claim

that

e-tailing

giant

Amazon.com's

high-profile

dissemination

of

used

books

eats

into

their

often

limited

royalties

and

shortens

the

period

they

have

to

generate

new

book

sales.

This

has

prompted

fears

that

publishers

will

be

forced

to

raise

the

prices

of

new

books,

in

order

to

extract

as

high

a

price

as

they

can,

during

the

onetime

sale

of

a

book.Amazon

has

argued

that

its

electronic

secondary

markets

actually

spur

new

goods

sales.

Complicating

this

argument

is

the

fact

that

Amazon

earns

about

the

same

from

selling

a

new

book

as

the

commission

it

generates

from

the

sale

of

a

used

book

on

its

marketplace

Thus

the

incentives

of

the

retailers

and

the

publishers

are

not

aligned.

In

fact,

the

commissions

generated

from

secondhand

goods

(such

as

books,

CDs,

DVDs,

and

packaged

software

products)

are

proving

to

be

a

money-maker

for

Amazon,

and

it

is

expanding

its

policy

of

paying

commissions

to

its

affiliates

to

include

the

sale

of

used

goods.A

salient

feature

of

these

information

goods

is

that,

once

bought,

many

consumers

derive

a

limited

utility

from

keeping

them

and

are

willing

to

resell

them

if

the

transaction

costs

are

negligible.

Prior

research

has

shown

that

IT-enabled

electronic

markets

reduce

transaction

costs

and

search

costs

For

buyers

and

sellers

alike,

electronic

networks

such

as

the

Internet

reduce

the

transaction

cost

of

finding

each

other

without

regard

to

geographical

constraints.

In

addition,

buyers

incur

low

search

costs

when

they

compare

prices

and

the

quality

attributes

of

new

and

used

goods.

Since

quality

EFFECT

OF

ELECTRONIC

SECONDARY

MARKETS

ON

THE

SUPPLY

CHAIN

depreciation

in

such

goods

is

usually

minimal,

consumers

find

it

increasingly

profitable,

or

less

costly,

to

buy

them.

Basically,

lower

prices

and

tangible

quality

attributes

make

them

perfectly

suitable

for

trading

on

such

electronic

secondary

markets.The

nature

of

durable

goods

markets

has

long

been

the

subject

of

active

research

in

economics

and

marketing.

However,

the

digital

economy

is

changing

the

nature

of

interactions

in

these

markets

For

example,

we

observe

the

concurrent

selling

of

new

and

used

goods,

with

offers

of

both

types

of

goods

being

retrieved

in

response

to

a

user

query

as

shown

in

Figure

In

this

setting,

there

is

a

strong

possibility

that

used

goods

can

cannibalize

the

sales

of

new

goods.Literature

Review

THE

DIFFICULTY

OF

MAINTAINING

THE

MONOPOLY

POWER

on

durable

goods

is

due

in

part

to

the

problem

of

time-inconsistency,

first

pointed

out

by

Coase

who

conjectured

that

if

a

firm

were

to

exploit

its

residual

demand

in

future

periods,

then

rational

consumers

would

anticipate

this

behavior

and

price

would

fall

to

the

competitive

level

The

interrelationship

between

the

markets

for

new

and

used

goods

was

They

argued

that

a

monopolist

can

maintain

market

power,

despite

an

inability

to

commit

to

future

production,

by

restricting

the

used

market.

Using

the

textbook

market

as

an

example,

Miller

suggests

that

the

opening

of

secondary

markets

will

force

publishers

to

increase

new

goods

prices

in

order

to

extract

the

maximum

possible

profit

from

the

onetime

sale

of

a

new

good.

Liebowitz

delineates

conditions

under

which

a

monopolist

will

produce

goods

with

a

lower

durability

than

under

competition.

Further

research

in

this

area

has

formalized

Coase's

conjecture

(e.g.,

Bulow

who

shows

that

a

monopolist

can

avoid

the

commitment

problem

by

leasing

as

opposed

to

selling)

and

pointed

out

conditions

under

which

it

does

not

hold.

For

example,

Bond

and

Samuelson

show

that

depreciation

reduces

the

monopolist's

incentive

to

cut

price.

Kahn

shows

a

similar

result

by

assuming

an

upward-sloping

marginal

cost

schedule

instead

of

constant

marginal

costs.

Rust

solves

for

consumer

behavior

and

prices

for

used

goods

in

a

durable

goods

market,

with

stochastic

depreciation.

The

main

argument

of

these

earlier

papers

is

that

the

secondhand

markets

need

not

hurt

the

manufacturer

because

they

will

anticipate

the

resale

value

of

their

product

and

thereby

increase

the

new

good

price

accordingly.

But

a

more

recent

stream

of

literature

has

argued

that

the

secondary

market

also

creates

a

substitution

effect

that

the

models

above

ignore.

The

substitution

effect

accrues

from

the

fact

that

new

goods

face

competition

from

used

goods.

Accordingly,

some

new

goods

consumers

will

shift

to

the

used

goods

market

because

of

such

a

substitution

effect.

Thus

the

prior

literature

highlights

two

countervailing

effects

of

secondary

markets.

On

one

hand,

used

versions

of

the

durable

are

viewed

as

substitutes

for

the

newly

produced

durable

good.

On

the

other

hand,

the

price

of

a

durable

good

is

the

present

discounted

value

of

the

benefits

stream

associated

with

that

good

such

as

a

resale.

Hence,

the

optimal

extent

of

any

type

of

interference

is

conditioned

by

the

relative

magnitude

of

these

two

effects.

An

important

aspect

of

the

theoretical

models

in

prior

literature

is

that

they

have

explicitly

accounted

for

transaction

costs

incurred

in

resales,

given

that

these

costs

are

significant

in

the

physical

world.

Anderson

and

Ginsburgh

show

the

monopolist

could

benefit

from

imposing

a

variable

transaction

fee

on

buyers

of

used

goods.

Waldman

and

Hendel

and

Lizzeri

show

that

firms

use

the

lease-only

policy

of

reducing

the

availability

of

used

goods.

Aron

and

Sundararajan

highlight

how

installed

base

affects

a

firm's

incentive

to

have

a

secondary

market.

Porter

and

Sattler

introduce

a

transaction

cost

for

sellers

of

secondhand

goods

and

show

that

the

volume

of

trade

increases

with

imperfect

substitutability.

For

a

complete

survey

of

the

literature

on

durable

goods,

see

Waldman

.Empirical

work

on

used

goods

markets

includes

a

stream

of

literature

looking

at

adverse

selection

in

those

markets

Genesove

finds

only

modest

evidence

of

adverse

selection

in

dealer

auction

markets

for

used

cars.

Ghose

et

al.

empirically

estimate

the

net

loss

to

publishers

due

to

the

cannibalization

effect

of

secondary

electronic

markets

on

new

book

sales.

One

major

contribution

of

this

paper

is

to

propose

an

alternate

rationale

as

to

how

secondary

markets

can

mitigate

such

losses

that

accrue

to

suppliers

of

information

goods

such

as

book

publishers,

record

labels,

EFFECT

OF

ELECTRONIC

SECONDARY

MARKETS

ON

THE

SUPPLY

CHAIN

and

movie

studios.

Although

the

sale

of

used

goods

is

allowed

under

copyright

law,

the

piracy

literature

provides

many

interesting

parallels

to

our

work.

Piracy

may

also

increase

legitimate

demand

by

enabling

the

producer

to

credibly

commit

to

not

reduce

its

price

in

the

future

One

key

difference

between

pirated

and

used

information

goods

is

the

nature

of

the

good

itself.

Whereas

most

pirated

goods

are

digital

in

nature,

information

goods

traded

on

secondary

markets

are

generally

tangible.

We

also

explore

the

role

of

used

good

commissions

and

discuss

associated

managerial

insights.

Specifically,

we

examine

the

following:

Does

the

sale

of

used

goods

ever

benefit

suppliers

when

they

cannot

accrue

any

direct

profits

from

secondary

market

sales?

What

is

the

effect

of

used

goods

on

new

good

prices

and

profits?

What

strategic

implications

does

the

used

good

commission

set

by

the

retailer

have

for

suppliers?Key

Results

PREVIOUS

RESEARCH

HAS

SUBSTANTIALLY

ADVANCED

our

understanding

of

how

durable

goods

markets

function.

In

contrast

to

products

such

as

automobiles,

we

consider

information

goods,

such

as

books,

CDs,

DVDs,

or

software

products,

where

transaction

frequency

is

much

higher

because

transaction

and

search

costs

are

virtually

absent

in

IT-enabled

electronic

secondary

markets.

The

success

of

electronic

secondary

markets

for

these

types

of

durable

goods,

such

as

those

hosted

by

Amazon

or

eBay,

motivates

the

need

to

go

beyond

existing

models.

Our

model

simultaneously

accommodates

commonly

observed

phenomena,

such

as

active

secondary

markets,

heterogeneity

in

consumer

preferences,

endogenous

demand

functions,

quality

depreciation,

and

the

degree

of

product

substitutability

between

competing

goods.

Before

discussing

our

model,

we

highlight

three

features

that

distinguish

our

work

from

prior

research

and

lead

to

our

main

contributions.First,

in

our

model,

we

consider

two

different

types

of

a

similar

good,

each

being

sold

by

two

different

suppliers

via

one

common

retailer.

There

exists

an

underlying

heterogeneity

in

consumers'

purchasing

affinity

for

both

goods.

Hence

the

presence

of

secondary

markets

not

only

expands

the

demand

for

a

specific

type

of

good

but

also

opens

up

the

possibility

of

some

consumers

purchasing

a

second

good

of

another

type.

The

"duopolistic"

nature

of

the

market

motivates

suppliers

to

compete

even

more

strongly

on

Further,

the

presence

of

a

secondary

market

allows

some

consumers

to

replace

their

used

goods

of

one

type

with

new

goods

of

the

other

type.

We

show

that

the

existence

of

the

secondary

market

leads

to

an

increase

in

their

valuation

for

the

new

good,

which

in

turn

can

lead

to

higher

supplier

profits.Second,

in

our

model,

suppliers

do

not

accrue

any

direct

gains

from

used

good

sales

and

hence

they

cannot

directly

use

secondhand

goods

for

practicing

intertemporal

price

discrimination.

The

retailer

that

acts

as

an

intermediary

in

our

model

sets

the

used

good

commission

and

captures

all

direct

secondary

market

benefits.

Consequently,

its

interests

are

not

aligned

with

those

of

the

supplier.

In

such

a

scenario,

the

used

good

commission

plays

a

strategic

role

by

determining

the

precise

change

in

consumers'

implicit

disposable

income

from

a

resale.

From

a

managerial

perspective,

GHOSE,

TELANG,

AND

KRISHNAN

we

show

that

despite

the

threat

of

cannibalization

of

new

goods

by

used

goods,

a

retailer

has

a

strong

incentive

to

establish

a

secondary

market.

While

this

is

always

detrimental

for

suppliers

in

a

monopoly,

a

duopolistic

secondary

market

leaves

them

better

off,

for

a

wide

range

of

parameters.

Further,

we

show

that

although

increasing

the

used

good

commission

affects

suppliers

favorably

in

a

monopoly,

it

affects

them

adversely

in

a

competitive

scenario.Third,

some

consumers

derive

a

positive

utility

from

holding

the

used

good.

This

feature,

along

with

the

duopolistic

market

structure

where

a

supplier

wants

to

sell

its

goods

in

a

competitor's

market,

enables

suppliers

to

decrease

their

prices

in

the

presence

of

a

secondary

market.

This

is

in

contrast

to

the

conventional

perspective

that

new

good

prices

tend

to

increase

in

the

presence

of

a

used

good

market.

Further,

we

point

out

that

irrespective

of

the

nature

of

the

market

(monopolistic

or

duopolistic),

an

increase

in

the

new

good

price

leads

to

an

increase

in

the

used

good

price.

Using

data

from

Amazon,

we

provide

empirical

evidence

that

corroborates

these

results.Firms

THE

MODEL

CONSISTS

OF

TWO

SUPPLIERS,

S

A

and

S

B

selling

goods

A

and

B,

respectively,

through

one

common

retailer,

to

a

unit

mass

of

consumers,

each

in

two

different

markets.

Suppliers

are

assumed

not

to

have

any

capacity

limits.

In

our

model,

one

can

think

of

the

two

suppliers

as

two

publishers

or

two

record

labels

selling

two

different

books

or

CDs,

respectively.

The

product

i

(i

A,

B)

that

is

sold

by

each

supplier

is

a

durable

that

provides

two

periods

of

service.

Thus

a

new

good

provides

service

for

two

periods,

irrespective

of

the

period

in

which

it

is

bought.

A

good

is

new

when

it

is

sold

in

period

and

the

same

good

is

classified

as

used

in

period

In

the

first

period,

only

new

goods

are

available.

In

the

second

period,

the

retailer

opens

a

used

good

market

where

consumers

can

buy

and

sell

used

goods

with

minimal

transaction

costs.

The

retailer

sells

both

the

goods,

A

and

B,

supplied

by

each

of

the

respective

suppliers.

Whenever

the

consumer

sells

a

used

good

(irrespective

of

whether

it

is

type

A

or

B),

the

retailer

gets

a

commission

k

U

(where

k

U

per

used

good

sold,

whereas

the

rest

-k

U

is

the

gain

to

the

consumer.

Similarly

it

gets

k

N

(where

k

N

per

new

good

sold,

with

the

rest

-k

N

going

to

the

supplier.

We

also

assume

that

the

marginal

cost

of

production

of

each

good

is

zero

for

the

suppliers.

This

is

true

for

a

variety

of

information

goods

such

as

books,

CDs,

DVDs,

videos,

journals,

shrink-wrapped

software,

and

so

on.Consumers

are

heterogeneous

in

their

valuations

of

the

good.

Let

θ

be

a

consumer's

marginal

valuation

for

the

quality

of

a

good,

where

θ

For

any

given

quality,

a

consumer

with

a

higher

θ

is

willing

to

pay

more

for

the

product

than

one

with

a

lower

θ.Buying

a

used

good

entails

a

cost

of

some

degree

of

quality

degradation.

Let

denote

the

quality

of

the

new

good

and

q

denote

the

quality

of

the

used

good

in

period

where

q

Thus

q

can

be

interpreted

as

the

degree

of

intertemporal

quality

degradation

of

the

new

good

over

two

periods.

If

a

consumer

purchases

a

product

of

quality

q

at

price

p,

his

or

her

utility

is

U(θ)

θq

-p.

That

is,

the

increase

in

utility

with

quality

is

higher

for

consumers

with

higher

valuations.In

our

model,

there

are

two

types

of

consumers.

Type

A

consumers

have

θ

valuation

for

good

A

and

αθ

valuation

for

good

B,

where

α

Similarly,

type

B

consumers

have

θ

valuation

for

good

B

and

αθ

valuation

for

good

A.

The

parameter

α

is

the

degree

of

consumers'

affinity

for

the

other

product

such

that

the

higher

the

value

of

α,

the

more

intense

is

the

competition

between

the

two

suppliers

for

inducing

the

consumers

to

buy

their

goods.

When

α

consumers

strictly

prefer

the

specific

good

i

(A,

B)

according

to

their

respective

types.

However,

α

implies

consumers

have

an

incentive

to

buy

both

goods.

In

this

paper,

we

refer

to

α

as

the

cross-product

purchase

affinity

parameter.Having

bought

a

new

good

in

period

consumers

can

either

hold

onto

the

good

or

sell

it

as

a

used

good

in

the

market

in

the

second

period.

Intuitively,

we

would

expect

that

for

the

same

θ,

a

consumer

holding

the

good

will

derive

less

utility

than

a

used

good

buyer.

Therefore,

we

assume

that

while

the

buyer

of

a

used

good

derives

a

utility

of

θq,

the

holder

of

the

used

good

derives

a

utility

of

θh,

where

h

q.

To

motivate

this

discussion,

consider

a

good

such

as

a

DVD

or

a

fiction

novel.

Once

a

consumer

has

seen

the

movie

or

read

the

novel,

his

or

her

utility

from

the

product

content

will

decline.

As

such,

he

or

she

will

derive

a

lower

utility

from

holding

the

good.

Conversely,

a

buyer

of

the

same

used

DVD

or

novel

can

derive

a

much

higher

utility

from

the

content

of

the

good.

Hence,

depending

on

the

prices

of

both

goods

and

consumers'

relative

preference

for

each

good

(α),

they

may

choose

to

buy

one

or

both

the

goods.

The

game

is

modeled

as

a

multistage

process

across

two

periods.

We

consider

a

subgame-perfect

equilibrium

of

this

game

using

backward

induction.WE

BEGIN

WITH

THE

BENCHMARK

CASE

when

there

is

no

market

for

used

goods.

This

allows

us

to

do

comparative

statics

when

an

active

used

good

market

exists.

We

consider

two

separate

Case

"Monopoly

Without

Used

Goods

Case."

In

this

case,

consumers

purchase,

at

most,

one

good

from

either

supplier,

according

to

their

type

i,

i

(A,

B).

Case

"Duopoly

Without

Used

Goods

Case."

In

this

case,

some

consumers

are

interested

in

purchasing

both

goods,

one

from

each

supplier

i,

i

(A,

B).Throughout

the

rest

of

the

paper,

we

refer

to

these

scenarios

as

the

"no

used

good

case"

for

both

monopolistic

and

duopolistic

markets.

We

solve

for

the

equilibrium

price

in

both

cases.

We

will

then

perform

a

similar

analysis

in

the

presence

of

a

used

GHOSE,

TELANG,

AND

KRISHNAN

good

market

and

do

some

comparative

statics.

These

comparisons

enable

us

to

highlight

the

effect

of

a

secondary

market

on

suppliers

and

the

retailer.Since

there

is

no

secondary

market,

consumers

can

only

do

the

following:

buy

the

new

good

(N)

in

the

first

period

and

hold

it

(H)

in

the

second

period.

In

the

second

period,

for

a

consumer

of

type

θ,

the

utility

derived

from

holding

it

is

simply

θh.

We

denote

the

price

of

a

new

good

in

this

scenario

as

P

N

mn

where

the

superscript

N

denotes

that

it

is

the

price

of

a

new

good

and

the

subscript

mn

denotes

that

it

is

the

"monopoly

without

used

goods

case."

A

consumer

of

type

θ

in

each

market

will

buy

a

good

as

long

aswhere

θ′

is

the

consumer

indifferent

to

buying

a

good

and

not

buying

at

all.

Here,

P

N

mn

can

be

interpreted

as

the

monopoly

price

that

S

A

(S

B

offers

to

all

consumers.

At

this

price,

the

demand

for

a

new

good

(both

A

and

B)

iswhereas

the

total

profit

of

the

retailer

from

both

the

markets

isNote

that

the

implicit

assumption

here

is

that

consumers

have

a

disposable

income

I

equal

to

at

least

the

price

of

one

new

good-that

is

h)k

N

be

able

to

purchase

the

good.For

all

α

both

suppliers

offer

the

monopoly

price

(P

N

mn

P

N

mn

and

consumers

buy,

at

most,

one

good

of

their

type.For

α

each

supplier

poaches

into

its

rival's

market

and

a

stiff

price

competition

Hence,

to

focus

on

the

implications

of

a

secondary

market

on

suppliers

and

the

retailer,

we

will

consider

the

case

of

α

in

the

subsequent

discussion

for

the

monopolistic

market.

It

is

also

worth

noting

that

if

consumers

did

not

derive

any

utility

from

holding

the

good,

then

the

optimal

price

P

N

mn

suppliers

could

have

charged

would

have

been

equal

to

The

increased

utility

from

holding

the

good

is

what

drives

suppliers

to

charge

the

extra

price

the

absence

of

a

secondary

market,

the

profit-maximizing

prices

for

the

retailer

and

the

suppliers

are

the

same.

However,

when

there

is

a

secondary

market,

the

strategies

for

the

retailer

and

suppliers

change,

because

although

the

retailer

still

gets

a

commission

for

each

used

good

sold,

the

suppliers

do

not.

It

is

this

differential

incen-tive

for

used

good

sales

between

suppliers

and

retailers

and

the

consequent

implications

for

equilibrium

profits

that

we

aim

to

explore

in

this

paper.Case

Duopoly

Without

Used

Goods

Now,

we

consider

the

case

when

some

consumers

can

purchase

both

goods.

That

is,

some

consumers

of

type

A

are

willing

to

also

buy

good

B

and

vice

versa.

In

the

previous

equilibrium,

we

showed

that

both

suppliers

simultaneously

offer

monopoly

prices

when

consumers

are

buying

one

good

according

to

their

type.

Clearly,

the

suppliers'

pricing

strategy

will

change

when

each

supplier

can

sell

its

good

in

the

other's

market

as

well.Let

the

subscript

dn

denote

the

duopoly

market

with

no

used

goods.

Suppose

S

A

offers

price

P

N

dn

such

that

in

a

type

B

market,

some

consumers

find

it

incentive

compatible

to

buy

both

good

A

and

B

at

lower

prices,

as

long

as

it

satisfies

their

individual

rationality

constraint.

Hence,

the

type

B

market

gets

split

into

two

segments,

such

that

the

higher-willingness-to-pay

consumers

buy

both

good

B

and

good

A,

whereas

the

remaining

buyers

buy

only

good

B.

It

follows

that

all

type

B

consumers

between

θ

and

also

buy

from

S

A

By

symmetry,

supplier

S

B

will

do

the

same

in

a

type

A

market

by

setting

a

price

P

N

dn

We

formally

show

that

there

exists

a

pure

strategy

equilibrium

in

prices.It

is

critical

to

note

that

in

order

for

consumers

to

buy

both

goods,

they

should

have

a

disposable

income

I

equal

to

at

least

twice

the

price

of

a

new

good-that

is,

α).

If

this

is

not

so,

then

they

can

buy

only

one

good,

which

will

be

sold

at

the

monopoly

price,

as

shown

in

the

last

section.

Next,

we

show

that

when

a

secondary

market

is

established,

the

disposable

income

required

to

buy

both

goods

declines

considerably,

because

consumers

are

able

to

trade

their

used

good

for

some

extra

income.

This

increases

their

valuation

for

a

new

good

and,

in

turn,

positively

affects

the

supplier's

and

retailer's

profitability.

Retailer

Establishes

an

Electronic

Secondary

Market

IN

THIS

SCENARIO,

CONSUMERS

ARE

AWARE

that

in

period

only

new

goods

are

available,

whereas

in

period

both

new

and

used

goods

are

available.

Consumers

who

purchase

the

new

good

in

period

always

have

the

option

of

selling

their

used

good

in

a

secondary

market

in

period

Thus

at

the

beginning

of

each

period,

when

consumers

evaluate

their

needs

over

the

two-period

horizon,

they

can

follow

one

of

five

independent

strategies

laid

out

below.

Let

P

U

mn

and

P

U

ms

denote

the

new

good

and

used

good

prices,

respectively,

where

the

superscript

U

indicates

the

presence

of

a

secondary

market.

The

subscripts

ms

and

mn

denote

a

monopolist

with

and

without

used

good

cases,

respectively.

GHOSE,

TELANG,

AND

KRISHNAN

Let

N

i

H,

S,

U,

and

I

denote

a

single-period

action,

corresponding

to

buying

a

new

good

of

type

i,

holding

onto

a

used

good,

selling

a

used

good,

buying

a

used

good,

and

remaining

inactive

by

not

buying

any

good.

We

state

a

general

formulation

of

underlying

utility

and

then

endogenously

derive

the

consumer

demand

functions.

Doing

so

enables

us

to

get

a

better

understanding

of

consumer

strategies

and

the

trade-offs

involved

in

choosing

among

the

differentiated

products.

Similar

to

the

previous

section,

we

will

again

focus

on

two

separate

Case

"Monopoly

with

Used

Goods

Case."

Consumers

are

interested

in

purchasing

at

most

one

good.

Case

"Duopoly

with

Used

Goods

Case."

Some

consumers

purchase

both

goods.Case

Monopoly

with

Used

GoodsWe

next

discuss

a

supplier's

problem

of

determining

the

optimal

new

good

price

when

the

retailer

establishes

a

used

good

market.

We

assume

the

used

market

is

competitive

and

neither

the

supplier

nor

the

retailer

has

any

direct

control

in

setting

the

used

good

price.

Hence,

a

used

good

sale

does

not

provide

any

explicit

benefit

to

the

suppliers.

When

consumers

buy,

at

most,

one

good,

they

buy

their

preferred

good

according

to

the

market

type

to

which

they

belong.

That

is,

consumers

of

type

A(B)

buy

good

A(B).

Therefore,

the

corresponding

utilities

derived

from

various

action

strategies

are

as

follows:

Figure

describes

the

segmentation

of

the

market

based

on

consumer

action

strategies.

Recall

that

in

this

case

a

consumer

is

interested

in

buying

at

most

one

good.

Note

that

the

condition

-k

U

h

needs

to

hold

for

such

a

market

segmentation.

Intuitively,

this

means

that

some

consumers

derive

a

higher

utility

from

selling

the

used

good

than

from

holding

onto

it.

Further,

the

holding

utility

also

needs

to

be

such

that

any

consumer

finds

it

incentive

compatible

to

hold

on

to

a

used

good,

instead

of

selling

it.

Using

the

individual-rationality

and

the

incentive-compatibility

constraints,

we

derive

the

three

indifferent

points

that

define

the

consumer

market

segments.

The

market

cutoffs

are

as

follows:Note

that

the

number

of

consumers

in

any

group

is

given

by

the

length

of

that

segment.

It

is

important

to

recognize

that,

in

our

model,

based

on

suppliers'

pricing

EFFECT

OF

ELECTRONIC

SECONDARY

MARKETS

ON

THE

SUPPLY

CHAIN

strategies,

the

number

of

consumers

in

these

groups

will

emerge

endogenously.

Since

the

used

good

market

is

competitive,

the

price

of

used

goods,

p

s

will

also

be

endogenously

determined.

This

ensures

that

the

market-clearing

conditions

equalize

supply

and

demand

of

used

goods.

All

consumers

from

θ

to

θ

are

suppliers

of

used

goods.

All

consumers

from

θ

θ

create

the

demand

for

used

goods.

By

equating

demand

with

supply,

we

get

the

market-clearing

used

good

price,

P

U

ms

The

optimal

new

good

price,

P

U*

mn

is

derived

from

the

supplier's

profit

equation,

after

substituting

for

P

which

gives

the

supplier's

profit

asand

the

retailer's

profit

as

The

presence

of

an

active

used

goods

market

creates

competition

for

new

goods

because

the

option

of

buying

used

goods

is

now

incentive-compatible

for

some

consumers

who

would

have

bought

new

goods

before.

This

enhanced

competition

forces

suppliers

to

decrease

new

good

prices

in

order

to

remain

competitive

with

used

goods.

Recall

that

the

supplier

cannot

derive

any

benefit

from

the

sale

of

used

goods,

since

all

the

used

goods

commission

is

forfeited

to

the

retailer.

In

contrast

to

conventional

wisdom,

we

find

that

the

price

of

a

new

good

decreases

from

the

case

when

there

is

no

used

good

market

to

when

there

is

an

active

used

good

market.

This

is

surprising,

but

the

intuition

accrues

from

the

fact

that

the

existence

of

the

NH

segment

discourages

suppliers

from

raising

prices.

Thus

we

point

out

that

as

long

as

some

consumers

derive

a

positive

utility

from

holding

a

used

good,

suppliers

will

not

raise

the

new

good

price,

despite

the

establishment

of

a

secondary

market

by

an

intermediating

retailer.On

the

other

hand,

if

we

compare

retailer

profits

in

both

cases,

we

observe

that,

in

general,

retailer

profits

increase

with

the

establishment

of

a

secondary

market.

The

loss

from

lower

profit

margins

per

customer

in

NH

and

NS

segments

is

more

than

offset

by

the

gain

from

additional

sales

in

the

IU

segment.

This

result

highlights

why

retailers

of

durable

goods

who

split

the

gains

from

used

good

sales

with

consumers

only

(not

with

suppliers)

have

a

strong

incentive

to

open

a

secondary

market.The

analysis

above

has

interesting

managerial

and

public

policy

implications.

First,

even

though

secondary

markets

leave

suppliers

worse

off,

in

general,

the

retailer

is

better

off

by

prudently

setting

the

used

good

commission

after

observing

the

supplier's

new

good

commission.

More

importantly,

consumers

are

always

better

off.

Therefore,

from

a

policy

perspective,

even

though

secondary

markets

are

detrimental

for

suppliers,

they

do

enhance

consumer

welfare.

Second,

the

analysis

allows

us

to

observe

the

effect

of

various

parameters

on

firms'

optimal

strategies.

For

example,

retailers

such

as

Amazon

will

be

interested

in

setting

an

optimal

commission

k

U

When

the

used

good

commission

k

U

increases,

the

used

good

market

shrinks

in

size

while

the

new

good

market

expands.

This

leads

to

higher

supplier

profits.

Although

this

tends

to

lower

profits

for

the

retailer

from

used

good

sales,

it

is

compensated

by

the

increased

profits

from

the

new

good

market.

But

importantly,

unlike

suppliers,

the

retailer

benefits

twice

from

the

NS

segment:

it

earns

the

commission

twice

in

the

form

of

k

N

and

k

U

Therefore,

with

a

higher

used

good

commission,

eventually,

the

used

good

market

shrinks

enough

to

start

decreasing

the

retailer's

net

profits.

Clearly,

then,

beyond

a

critical

value

of

k

U

the

retailer's

incentive

to

increase

k

U

is

at

odds

with

that

of

the

suppliers,

who

always

prefer

a

shallow

used

good

market.

Formally,

the

optimal

k

U

for

the

retailer

is

shown

below.(ii)

Further,

as

the

supplier's

new

good

commission

k

N

increases,

the

retailer's

optimal

used

good

commission

k

U

also

increases.Intuitively,

a

higher

k

N

implies

that

the

retailer

generates

more

profit

from

the

new

good

market.

Hence,

its

best

response

is

to

minimize

the

size

of

the

used

good

market

by

increasing

k

U

thereby

making

it

less

profitable

for

consumers

to

sell

the

used

good.

Basically,

by

strategically

increasing

k

U

the

retailer

can

alter

the

size

of

the

NH

segment

vis-à-vis

the

NS

segment

and

thus

make

higher

This

analysis

highlights

the

effect

of

a

secondary

market

on

a

monopolistic

supplier,

and

a

retailer's

profits,

as

well

as

on

consumer

welfare.

One

key

contribution

is

to

analyze

the

competitive

implications

when

some

consumers

can

buy

both

goods.

We

next

analyze

the

effect

of

a

duopolistic

used

good

market,

wherein

some

consumers

purchase

both

goods.We

are

looking

for

two

main

insights.

First,

what

is

the

effect

on

suppliers'

profits

when

they

are

able

to

sell

goods

in

their

competitors'

market

as

well?

Second,

what

will

be

the

required

disposable

income

for

consumers

to

buy

both

goods?

In

this

section,

we

proceed

to

answer

these

questions

by

introducing

the

element

of

intersupplier

competition

in

the

presence

of

a

used

good

market,

because

some

consumers

are

willing

to

buy

two

goods.We

continue

with

the

stylized

scenario

of

two

symmetric

suppliers,

selling

in

two

different

markets

of

equal

size,

reduced

to

a

unit

mass

of

one.

Based

on

the

strategies

outlined

below,

consumers

are

segmented

into

various

consumption

classes

according

to

their

willingness

to

pay

for

the

good.

Recall

that

in

this

case,

some

consumers

are

interested

in

buying

two

goods,

one

from

each

supplier,

as

long

as

it

gives

them

a

positive

surplus.

This

leads

to

the

creation

of

a

new

segment,

which

we

term

NSN.

The

dominant

consumer

strategies

and

corresponding

consumer

utility

functions

are

shown

below.

Similar

to

the

monopoly

case,

by

equating

the

incentive-compatibility

constraints

based

on

these

four

strategies,

we

derive

the

three

indifference

points

that

define

the

consumer

segments

in

each

market,

i

or

j,

i,

j

(A,

B),

which

are

as

follows:Note

that

as

before,

the

grouping

of

consumers

outlined

above

corresponds

directly

to

their

overall

valuations

of

the

good.

That

is,

consumers

with

higher

valuations

for

GHOSE,

TELANG,

AND

KRISHNAN

the

good

will

tend

to

be

on

the

right,

followed

by

consumers

with

lower

valuations

in

the

middle,

and

so

on

toward

the

left.

This

is

shown

in

Figure

The

class

division

points

are

derived

from

the

individual-rationality

and

the

incentive-compatibility

constraints.

Consumers

in

follow

IU

i

and

those

in

(θ

i

will

follow

II.

All

consumers

from

to

θ

i

are

the

suppliers

of

used

goods.

All

consumers

from

(θ

i

θ

i

create

the

demand

for

used

goods.

By

equating

the

demand

of

used

goods

with

the

supply,

we

get

the

market-clearing

second-period

price

of

used

goods

P

U

i

ds

The

optimal

new

good

price,

P

U*

i

dn

can

be

derived

from

the

supplier's

profit

equation,

after

substituting

for

P

U

i

ds

Note

that

new

good

sales

occur

in

the

first

two

segments,

that

is,

in

the

segments

labeled

N

i

SN

j

and

N

i

H.

Also,

since

consumers

in

N

i

SN

j

segment

buy

from

both

the

suppliers,

each

supplier

also

gets

the

N

j

SN

i

segment

of

the

competitor's

market.

Hence,

the

supplier's

profit

is

given

byThis

yields

the

optimal

new

good

price

asThe

expressions

for

optimal

used

price

and

supplier

and

retailer

profits

are

given

in

the

Appendix.

Before

we

perform

comparative

statics,

the

reader

should

note

that

the

new

good

prices

and

profits

are

increasing

in

α.

This

is

because

a

higher

α

implies

that

the

N

i

SN

j

segment

is

willing

to

pay

more

for

the

good.

Formally,

we

show

the

following

proposition:The

new

good

prices

increase

monotonically

with

α,

the

consumers'

degree

of

preference

for

the

other

good,

but

decrease

with

the

used

good

commission,

k

U

Further,

used

good

prices

are

increasing

with

the

new

good

prices.In

a

monopolistic

scenario,

suppliers

prefer

a

high

k

U

because

this

leads

to

a

smaller

used

good

market,

which

suits

their

interests.

On

the

other

hand,

when

some

highvaluation

customers

are

willing

to

buy

both

goods,

suppliers

prefer

to

have

an

active

used

good

market.

This

is

because

the

existence

of

the

NSN

segment

critically

depends

on

the

ability

of

some

consumers

to

sell

the

used

good

and

extract

the

extra

income

from

the

resale.

Proposition

shows

that

k

U

has

an

adverse

effect

on

new

good

prices

and

profits.

This

is

because

with

a

higher

k

U

consumers

get

a

reduced

benefit

from

selling

the

used

good

compared

to

holding

it.

Hence

the

NH

segment

increases

in

size

whereas

the

NSN

segment

shrinks.

To

compensate

for

the

lower

net

demand

of

new

goods,

the

suppliers

are

forced

to

reduce

their

prices

to

sustain

the

NSN

segment,

which,

for

the

most

part,

reduces

their

profits.

Therefore,

unlike

the

monopoly

case,

the

suppliers

now

prefer

a

lower

k

U

On

the

other

hand,

the

retailer

is

interested

in

setting

a

higher

k

U

in

order

to

extract

all

the

benefits

from

used

good

sales.

Again,

this

highlights

that

retailers

and

suppliers

are

incompatible

in

their

incentives

in

selecting

an

optimal

k

U

Next,

we

will

show

that

when

the

secondary

market

is

modeled

as

a

duopolistic

market,

suppliers

are

not

necessarily

adversely

affected.

Moreover,

new

good

prices

are

generally

lower

in

such

markets.Duopoly

with

Used

Good

Versus

Duopoly

Without

Used

Goods

ALTHOUGH

WE

GET

CLOSED-FORM

SOLUTIONS

for

the

price

and

profit

expressions,

the

highly

nonlinear

equations

do

not

yield

easy

analytical

comparisons.

Therefore,

we

use

numerical

analysis

to

gain

some

insight.

First,

we

point

out

the

following:

In

other

words,

when

some

consumers

buy

both

goods,

then

there

exists

an

α

m

such

that

for

α

α

m

suppliers

are

better

off

in

the

presence

of

a

used

good

market.

To

see

this,

first

note

that

for

low

α,

while

the

prices

and

profits

in

a

duopolistic

market

without

used

goods

are

very

low,

this

is

not

necessarily

true

in

a

duopolistic

used

good

market.

The

reason

is

that

the

used

good

market

allows

consumers

to

sell

their

goods,

which

increases

their

valuation

for

a

new

good.

Hence,

suppliers

can

extract

this

increased

valuation

by

charging

somewhat

higher

prices

than

in

the

case

without

a

used

good

market.

This

is

similar

to

the

notion

of

"indirect

appropriability,"

as

pointed

out

in

the

prior

literature.

We

note

thatfor

low

α-that

is,

suppliers

are

better

off

with

a

used

good

market

for

lower

values

of

α.

Conversely,

for

higher

values

of

α,

we

find

thatClearly,

if

the

difference

in

the

profit

function

under

the

two

scenarios

were

positive

and

monotonically

increasing,

then

that

would

imply

the

existence

of

an

α

m

such

that

suppliers

are

better

off

in

the

presence

of

a

used

good

market

for

all

α

α

m

Since

it

is

difficult

to

show

this

analytically,

we

use

numerical

simulations

to

determine

the

values

of

α

m

.GHOSE,

TELANG,

AND

KRISHNANFor

all

α

[α

n

suppliers'

optimal

new

good

price,

P

U

dn

in

the

duopolistic

used

good

market

is

lower

than

the

price

in

the

duopolistic

no

used

good

market,

P

N

dn

We

find

that

for

a

wide

range

of

α,

the

used

goods

marketplace

actually

leads

to

a

decrease

in

the

price

of

new

goods,

rather

than

higher

new

good

prices.

One

would

expect

that,

since

suppliers

do

not

gain

directly

from

the

sale

of

used

goods,

they

would

necessarily

increase

the

new

good

prices

to

extract

the

maximum

surplus

from

the

onetime

sale,

as

previous

research

has

noted

Basically

consumers'

affinity

for

the

second

good

induces

suppliers

to

lower

the

price

of

the

new

good

to

gain

additional

sales

in

the

competing

suppliers'

market.

By

lowering

their

prices,

suppliers

can

now

sustain

and

expand

the

NSN

market

segment,

thereby

accruing

a

higher

profit.

We

term

this

the

market

expansion

effect.

This

brings

down

the

new

good

prices

for

a

wide

range

of

parameters.Next,

we

compare

the

monopolistic

used

good

market

with

the

duopolistic

one.For

all

α

[α

dm

the

suppliers'

optimal

new

good

price,

P

U

dn

in

the

duopolistic

used

good

market

is

higher

than

the

price

in

the

monopolistic

used

good

market,

P

U

mn

Similarly,

we

point

out

that

for

some

α

[α

c

suppliers'

profits,

Π

d

U

(S),

in

a

duopolistic

used

good

market

are

higher

than

the

monopolistic

used

good

market

profits,

Π

m

U

(S).

We

know

that

at

very

low

values

of

α,

the

new

good

price,

P

U

dn

and

profits

in

a

duopolistic

market

are

very

low.

But

the

monopolistic

price,

P

U

mn

and

profits

do

not

depend

on

α,

and

hence

they

are

higher.

Moreover,

recall

from

the

previous

section

that

in

the

presence

of

a

duopolistic

used

good

market,

the

new

good

price,

P

U

dn

and

profits,

Π

d

U

(S),

are

increasing

in

α.

Therefore,

there

exists

an

α

dm

such

that

for

α

[α

dm

P

U

dn

P

U

mn

and

an

α

c

such

that

for

α

[α

c

P

d

U

(S)

P

m

U

(S).

Basically,

intersupplier

competition

keeps

the

prices

down,

whereas

the

increased

valuation

keeps

the

demand

up.

The

NSN

segment

boosts

the

demand

to

a

level

such

that

the

market

expansion

effect

dominates

the

price

competition

effect,

leaving

suppliers

better

off

even

under

the

duopolistic

market

structure.

In

summary,

this

section

highlights

the

implication

of

a

duopolistic

used

good

market

vis-à-vis

a

monopolistic

one

(which

has

been

commonly

studied

in

literature).

The

important

economic

effect

of

competition

in

a

secondary

market

is

that

it

creates

a

segment

of

consumers,

N

i

SN

j

who

are

willing

to

buy

their

less

preferred,

but

additional,

new

good

due

to

their

increased

valuation

for

a

good.

Segmenting

the

market

lowers

the

average

cost

of

owning

(or

renting)

a

good,

thus

creating

more

buyers.

In

effect,

the

used

good

option

is

like

a

rebate

coupon

that

reduces

the

net

price

consumers

actually

pay

for

the

new

good.

The

purchase

price

of

a

new

good

effectively

becomes

equal

to

the

discounted

sum

of

two

prices-the

new

good

price

in

period

minus

the

expected

used

good

price

in

period

Evidence

OUR

ANALYTICAL

MODEL

PROVIDES

us

with

two

testable

assertions.

First,

we

show

that

an

increase

in

the

availability

of

used

goods

leads

to

a

decrease

in

the

new

good

price.

Prior

work

and

conventional

wisdom

suggest

that

opening

a

used

good

market

would

lead

to

an

increase

in

new

good

prices.

Irrespective

of

the

value

of

the

holding

utility,

h,

once

the

effect

of

duopolistic

competition

is

taken

into

account,

new

good

prices

need

not

increase

with

the

establishment

of

secondary

markets.

Second,

we

show

that

the

used

good

price

increases

with

the

new

good

price.The

online

book

market

provides

a

rich

arena

to

test

the

robustness

of

these

two

results.

It

is

one

of

the

fastest

growing

industries

on

the

Internet.

Online

book

sales

grew

from

essentially

nothing

in

to

more

than

billion

in

Today,

such

sales

make

up

between

percent

and

percent

of

total

book

sales

in

the

United

States.

Our

data

set

is

compiled

from

publicly

available

information

on

new

and

used

book

prices

on

Amazon.com.

The

data

are

gathered

using

automated

Java

bots

to

download

and

parse

HTML

pages

from

Amazon.The

data

were

collected

in

two

separate

samples.

The

first

was

collected

from

September

to

March

and

the

second

between

April

and

July

Our

total

data

set

includes

approximately

individual

book

titles.

This

panel

of

books

includes

an

equal

number

of

books

from

each

of

five

major

categories:

New

York

Times

(NYT)

best

sellers,

former

NYT

best

sellers,

Amazon

best-seller

computer

books,

bestselling

textbooks,

and

"new

and

upcoming

books."

Current

best

sellers

were

included

because

of

their

popularity,

high

sales,

and

their

relevance

to

publisher

price

discrimination

strategies

(using

phased

releases

of

binding

types).

We

randomly

selected

books

appearing

in

the

NYT

best-sellers

list.

Using

the

LexisNexis

database,

we

included

a

sample

of

NYT

best

sellers

from

to

analyze

the

effect

of

time

on

the

availability

and

pricing

of

used

books.

Former

best

sellers

should

have

a

larger

base

of

copies

in

circulation

than

other

titles,

potentially

increasing

the

number

of

used

copies

for

sale.

Best-selling

computer

books

represent

one

of

the

most

popular

book

categories

sold

online,

and

are

also

subject

to

demand

shocks

based

on

the

version

updates

over

time

and

changes

in

technology.

Our

best-selling

computer

books

were

selected

at

random

from

the

best-seller

list

at

Amazon.

Textbooks,

which

are

also

included

in

our

survey,

have

similar

demand

characteristics

to

computer

books-in

terms

both

of

popularity

and

demand

shocks.

Our

textbooks

are

selected

at

random

from

facultyonline.com

best

sellers.

Our

final

category

is

new

and

upcoming

books.

This

category

allows

us

to

examine

the

development

of

used

book

markets

for

recent

releases

over

a

period

of

time.

Our

total

data

sample

includes

observations.For

each

of

the

books

we

tracked

that

is

uniquely

identified

by

an

International

Standard

Book

Number

(ISBN),

we

collect

data

on

new

book

prices

charged

by

Amazon,

as

well

as

new

and

used

book

prices

among

Amazon

marketplace

sellers.

Besides

prices,

for

each

offer

listed,

we

collect

the

book's

self-reported

condition,

and

the

seller's

marketplace

rating

(a

star

rating

provided

by

prior

customers).

Our

control

variables

include

the

log

of

the

time

since

the

book

was

released

(Datediff),

the

condition

of

the

lowest-priced

used

book

(Condition),

the

seller

rating

for

the

GHOSE,

TELANG,

AND

KRISHNAN

lowest-priced

used

book

(SellerRating),

and

the

log

of

the

number

of

used

books

offered

for

sale

for

a

particular

book

(CountofUsed).

Since

Amazon

provides

four

different

conditions

(or

quality

levels)

of

used

books

(like

new,

very

good,

good,

and

acceptable),

our

data

include

all

used

book

offers

on

a

given

date

for

each

condition.

Clearly,

since

including

each

used

book

offer

in

our

regression

is

impractical,

from

the

set

of

offers

we

selected

the

minimum

price

for

each

of

these

four

possible

conditions

of

a

given

used

book.

This

was

done

because

the

minimum

price

for

a

specific

condition

of

a

used

book

strictly

dominates

any

other

price

offers

at

that

condition.

Thus

we

derived

four

new

variables

constituting

the

minimum

prices

of

used

books,

denoted

by

P

Used

i(i

in

Table

which

contains

the

summary

statistics

for

our

data.

Formally,

we

use

the

following

terminology:

for

a

book

of

quality

"like

new,"

for

quality

"very

good,"

for

quality

"good,"

and

for

a

book

of

"acceptable"

quality.

Finally,

we

also

derived

the

minimum

price

on

a

given

day

for

all

offered

used

books

(across

all

conditions)

and

denote

it

as

P

UsedMin

We

denote

the

quality

of

a

used

book

by

the

variable

Condition.

Note

that,

given

the

manner

in

which

we

code

the

Condition

variable,

an

increase

in

condition

denotes

a

decrease

in

the

used

book

quality.

Table

lists

summary

statistics

for

our

data.

All

prices

listed

are

the

lowest

prices

for

each

category

for

each

ISBN.

We

did

this

because

many

of

the

used

and

new

nonAmazon

prices

in

our

data

set

were

substantially

higher

than

the

lowest

price

for

the

same

condition

listed

for

the

same

book

on

the

same

date.

Supplier

profit

in

a

monopoly

with

used

goods.Supplier

profit

in

a

duopoly

with

used

goods.Retailer

profit

in

a

monopoly

without

used

goods.Retailer

profit

in

a

duopoly

without

used

goods.Retailer

profit

in

a

monopoly

with

used

goods.Retailer

profit

in

a

duopoly

with

used

goods.Given

the

nature

of

our

data

(cross-sectional

plus

time

series)

we

use

a

fixed-effects

regression

model

to

test

our

hypothesis.

The

second

hypothesis

claims

that,

given

the

presence

of

used

books,

used

book

prices

are

higher

when

new

books

are

priced

higher.

To

alleviate

any

endogeneity

concerns,

we

ran

the

following

regression

using

lagged

values

(by

time

period).

The

variables

are

indexed

by

book,

b,

and

time,

t.

In

Equation

the

independent

variables

are

a

vector

of

"new"

non-Amazon

marketplace

prices

P

a

vector

of

Amazon

marketplace

prices

P

UsedMin

a

vector

of

the

number

of

used

goods

available,

and

a

vector

of

other

control

variables

(X).

In

Equation

the

independent

variables

are

Amazon

price

P

AN

a

vector

of

"new"

non-Amazon

marketplace

prices

P

a

vector

of

the

number

of

used

goods

available,

and

a

vector

of

other

control

variables

(X).

Since

we

estimate

a

fixed-effects

model,

book-specific

idiosyncrasies

are

captured

in

the

fixed-effect

constant.

The

estimates

are

presented

in

Table

.The

data

support

Hypotheses

and

In

Equation

the

coefficient

of

the

variable,

CountofUsed

is

negative

and

significant,

indicating

that

the

increase

in

the

availability

of

a

used

book

leads

to

lower

new

book

prices.

Further,

in

Equation

the

coefficient

of

the

term

P

AN

is

positive,

implying

that

if

the

new

book

price

increases,

then

the

used

book

price

also

increases.THE

ONGOING

HEATED

DEBATE

BETWEEN

suppliers

and

retailers

has

focused

on

the

supposed

damage

that

secondary

markets

established

by

online

retailers

are

inflicting

on

royalty

payments

and

supplier

profits.

Amazon's

secondary

market

is

flourishing,

with

more

consumers

discovering

that

buying

used

information

goods,

such

as

books,

CDs,

videos,

and

DVDs,

could

lead

to

significant

cost

savings.

However,

the

general

consensus

is

that

used

good

sales

cannibalize

new

good

sales

and,

consequently,

are

harmful

to

suppliers,

thereby

ruling

out

the

possibility

of

harmonious

coexistence

for

all

traditional

players

in

these

markets.Using

a

game-theoretic

model,

we

investigate

the

competitive

implications

of

these

newly

emerging

secondary

markets

on

supply-chain

profits

and

new

good

prices.

We

show

that

the

major

motivation

of

the

retailer

to

establish

a

used

good

market

is

to

capture

additional

surplus

from

those

consumers

who

were

unable

to

buy

in

the

new

good

market.

This

enables

the

retailer

to

effectively

practice

quality-based

price

discrimination

at

no

additional

cost.

Our

model

also

highlights

that,

unlike

a

monopolistic

market,

intersupplier

competition

can

be

beneficial

to

suppliers

in

the

presence

of

a

secondary

market,

rather

than

harmful.

This

occurs

because

the

presence

of

used

goods

subdues

the

rate

at

which

suppliers

can

increase

the

new

good

prices

with

consumers'

cross-product

purchase

affinity.

This

factor

affects

the

market

in

such

a

EFFECT

OF

ELECTRONIC

SECONDARY

MARKETS

ON

THE

SUPPLY

CHAIN

way

that

the

market

expansion

effect

compensates

for

the

price

competition

effect.

Thus,

we

show

that

contrary

to

popular

perceptions,

the

presence

of

a

used

good

market

is

beneficial

for

suppliers,

too,

under

a

relatively

wide

range

of

conditions.

When

merchandized

as

a

deep

discount

option

to

an

otherwise

inflated

new

good

price,

electronic

secondary

markets

offer

consumers

an

extremely

attractive

option

to

increase

their

valuation

for

a

new

good,

thereby

spurring

additional

new

good

sales.In

sum,

the

insights

from

this

model

help

to

explain

several

interesting

market

phenomena,

including

the

profitability

of

concurrent

sales

of

new

and

used

goods

for

suppliers-that

is,

why

secondary

markets

may

not

be

detrimental

for

them

if

one

considers

the

market

expansion

effect;

the

possibility

of

lower

new

good

prices

when

there

are

secondary

markets,

under

both

monopolistic

and

competitive

scenarios;

and

the

strategic

role

of

used

goods

commission

fees

and

their

differential

impact

on

equilibrium

prices

and

profits

for

suppliers

and

retailers,

under

monopolistic

and

duopolistic

markets.

Using

data

from

Amazon,

we

test

two

of

our

assertions

from

the

analytic

model

and

also

provide

empirical

estimates

of

the

effect

of

an

electronic

secondary

market

on

new

book

prices.Our

analysis

has

implications

for

suppliers

who

sell

through

intermediaries

but

cannot

appropriate

direct

gains

from

secondary

markets.

As

long

as

there

is

more

than

one

supplier,

and

consumers

are

willing

to

buy

another

new

good,

a

secondary

market

can

be

beneficial

for

suppliers.

Thus,

there

may

be

significant

gains

in

producer

welfare

from

the

additional

sales.

Indeed,

retailers,

such

as

Amazon,

and

suppliers

such

as

book

publishers

as

well

as

movie

studios

and

record

labels,

stand

to

benefit

and

earn

a

slice

of

the

growing

pie

created

by

lower

search

and

transaction

costs

of

electronic

secondary

markets.

As

more

consumers

migrate

online

and

adopt

such

resale

practices,

this

indirect

market

expansion

effect

can

further

alleviate

the

damages

incurred

by

suppliers

from

the

direct

effect

of

cannibalization

of

new

goods

by

used

goods.Another

implication

from

our

model

is

that

while

deriving

their

optimal

pricing

strategies,

suppliers

need

to

keep

in

mind

the

effect

of

a

positive

holding

utility

that

some

consumers

derive

from

keeping

their

used

goods,

instead

of

necessarily

selling

them.

This

is

especially

true

in

the

case

of

music

CDs

or

textbooks,

where

consumers

are

likely

to

listen

to

a

song

or

read

the

book

repeatedly,

compared

to

a

fiction

book

or

a

DVD

whose

utility

diminishes

at

a

much

faster

rate.

Ignoring

this

aspect

of

consumer

behavior

will

erroneously

drive

up

the

price

beyond

the

optimal

value.

Further,

this

caveat

also

holds

true

for

a

retailer's

deriving

its

optimal

used

good

commission,

because

the

extent

to

which

a

retailer

benefits

from

establishing

a

secondary

market

is

dependent

on

it.

As

pointed

out

earlier,

the

retailer's

optimal

used

good

commission

is

a

function

of

the

extent

of

quality

degradation

and

the

value

that

buyers

derive

from

holding

the

used

good

through

the

course

of

its

life.

It

is

conceivable

that

these

parameters

vary

among

information

goods,

and

the

broader

context

of

durable

goods

in

general.

So,

careful

consideration

needs

to

be

given

by

the

retailer

while

fixing

its

commissions

due

to

its

strong

effect

on

consumers'

buying

and

selling

patterns.

From

the

supplier's

perspective,

the

used

good

commission

is

critical

due

to

its

interdependency

with

the

new

good

commission.One

limitation

of

our

model

is

that

we

do

not

consider

more

than

two

suppliers

and

do

not

consider

an

n-period

model.

However,

our

intuition

is

that

the

main

effects

of

cross-product

purchase

affinity

(α)

driving

our

results

on

prices

and

profits

carry

through

with

multiple

suppliers

or

multiple

periods.

A

preliminary

analysis

reveals

that

incorporating

multiple

suppliers

or

multiple

resale

periods

restricts

the

space

of

α

only

where

the

supplier

benefits

from

used

good

markets.

Another

interesting

extension

would

be

to

incorporate

consumer

heterogeneity

in

a

second

dimension,

such

as

by

introducing

a

distribution

for

the

quality

degradation

parameter,

q.

In

such

a

scenario,

rather

than

all

used

good

sellers

selling

goods

at

the

same

quality,

q,

there

would

be

a

distribution

that

characterizes

the

seller-specific

degradation

of

the

quality

of

the

good.

We

hope

that

our

study

inspires

more

interest

and

paves

the

way

for

these

future

explorations.Acknowledgments:

The

authors

are

grateful

to

Uday

Rajan

for

insightful

feedback

and

Rob

Kauffman

for

detailed

suggestions.

They

thank

Erik

Brynjolfsson,

Anthony

Dukes,

seminar

participants

at

Carnegie

Mellon

University,

and

participants

at

THE

ONGOING

HEATED

DEBATE

BETWEEN

suppliers

and

retailers

has

focused

on

the

supposed

damage

that

secondary

markets

established

by

online

retailers

are

inflicting

on

royalty

payments

and

supplier

profits.

Amazon's

secondary

market

is

flourishing,

with

more

consumers

discovering

that

buying

used

information

goods,

such

as

books,

CDs,

videos,

and

DVDs,

could

lead

to

significant

cost

savings.

However,

the

general

consensus

is

that

used

good

sales

cannibalize

new

good

sales

and,

consequently,

are

harmful

to

suppliers,

thereby

ruling

out

the

possibility

of

harmonious

coexistence

for

all

traditional

players

in

these

markets..

Using

a

game-theoretic

model,

we

investigate

the

competitive

implications

of

these

newly

emerging

secondary

markets

on

supply-chain

profits

and

new

good

prices.

We

show

that

the

major

motivation

of

the

retailer

to

establish

a

used

good

market

is

to

capture

additional

surplus

from

those

consumers

who

were

unable

to

buy

in

the

new

good

market.

This

enables

the

retailer

to

effectively

practice

quality-based

price

discrimination

at

no

additional

cost.

Our

model

also

highlights

that,

unlike

a

monopolistic

market,

intersupplier

competition

can

be

beneficial

to

suppliers

in

the

presence

of

a

secondary

market,

rather

than

harmful.

This

occurs

because

the

presence

of

used

goods

subdues

the

rate

at

which

suppliers

can

increase

the

new

good

prices

with

consumers'

cross-product

purchase

affinity.

This

factor

affects

the

market

in

such

a

way

that

the

market

expansion

effect

compensates

for

the

price

competition

effect.

Thus,

we

show

that

contrary

to

popular

perceptions,

the

presence

of

a

used

good

market

is

beneficial

for

suppliers,

too,

under

a

relatively

wide

range

of

conditions.

When

merchandized

as

a

deep

discount

option

to

an

otherwise

inflated

new

good

price,

electronic

secondary

markets

offer

consumers

an

extremely

attractive

option

to

increase

their

valuation

for

a

new

good,

thereby

spurring

additional

new

good

sales..

In

sum,

the

insights

from

this

model

help

to

explain

several

interesting

market

phenomena,

including

the

profitability

of

concurrent

sales

of

new

and

used

goods

for

suppliers-that

is,

why

secondary

markets

may

not

be

detrimental

for

them

if

one

considers

the

market

expansion

effect;

the

possibility

of

lower

new

good

prices

when

there

are

secondary

markets,

under

both

monopolistic

and

competitive

scenarios;

and

the

strategic

role

of

used

goods

commission

fees

and

their

differential

impact

on

equilibrium

prices

and

profits

for

suppliers

and

retailers,

under

monopolistic

and

duopolistic

markets.

Using

data

from

Amazon,

we

test

two

of

our

assertions

from

the

analytic

model

and

also

provide

empirical

estimates

of

the

effect

of

an

electronic

secondary

market

on

new

book

prices..

Our

analysis

has

implications

for

suppliers

who

sell

through

intermediaries

but

cannot

appropriate

direct

gains

from

secondary

markets.

As

long

as

there

is

more

than

one

supplier,

and

consumers

are

willing

to

buy

another

new

good,

a

secondary

market

can

be

beneficial

for

suppliers.

Thus,

there

may

be

significant

gains

in

producer

welfare

from

the

additional

sales.

Indeed,

retailers,

such

as

Amazon,

and

suppliers

such

as

book

publishers

as

well

as

movie

studios

and

record

labels,

stand

to

benefit

and

earn

a

slice

of

the

growing

pie

created

by

lower

search

and

transaction

costs

of

electronic

secondary

markets.

As

more

consumers

migrate

online

and

adopt

such

resale

practices,

this

indirect

market

expansion

effect

can

further

alleviate

the

damages

incurred

by

suppliers

from

the

direct

effect

of

cannibalization

of

new

goods

by

used

goods..

Another

implication

from

our

model

is

that

while

deriving

their

optimal

pricing

strategies,

suppliers

need

to

keep

in

mind

the

effect

of

a

positive

holding

utility

that

some

consumers

derive

from

keeping

their

used

goods,

instead

of

necessarily

selling

them.

This

is

especially

true

in

the

case

of

music

CDs

or

textbooks,

where

consumers

are

likely

to

listen

to

a

song

or

read

the

book

repeatedly,

compared

to

a

fiction

book

or

a

DVD

whose

utility

diminishes

at

a

much

faster

rate.

Ignoring

this

aspect

of

consumer

behavior

will

erroneously

drive

up

the

price

beyond

the

optimal

value.

Further,

this

caveat

also

holds

true

for

a

retailer's

deriving

its

optimal

used

good

commission,

because

the

extent

to

which

a

retailer

benefits

from

establishing

a

secondary

market

is

dependent

on

it.

As

pointed

out

earlier,

the

retailer's

optimal

used

good

commission

is

a

function

of

the

extent

of

quality

degradation

and

the

value

that

buyers

derive

from

holding

the

used

good

through

the

course

of

its

life.

It

is

conceivable

that

these

parameters

vary

among

information

goods,

and

the

broader

context

of

durable

goods

in

general.

So,

careful

consideration

needs

to

be

given

by

the

retailer

while

fixing

its

commissions

due

to

its

strong

effect

on

consumers'

buying

and

selling

patterns.

From

the

supplier's

perspective,

the

used

good

commission

is

critical

due

to

its

interdependency

with

the

new

good

commission..

One

limitation

of

our

model

is

that

we

do

not

consider

more

than

two

suppliers

and

do

not

consider

an

n-period

model.

However,

our

intuition

is

that

the

main

effects

of

cross-product

purchase

affinity

(α)

driving

our

results

on

prices

and

profits

carry

through

with

multiple

suppliers

or

multiple

periods.

A

preliminary

analysis

reveals

that

incorporating

multiple

suppliers

or

multiple

resale

periods

restricts

the

space

of

α

only

where

the

supplier

benefits

from

used

good

markets.

Another

interesting

extension

would

be

to

incorporate

consumer

heterogeneity

in

a

second

dimension,

such

as

by

introducing

a

distribution

for

the

quality

degradation

parameter,

q.

In

such

a

scenario,

rather

than

all

used

good

sellers

selling

goods

at

the

same

quality,

q,

there

would

be

a

distribution

that

characterizes

the

seller-specific

degradation

of

the

quality

of

the

good.

We

hope

that

our

study

inspires

more

interest

and

paves

the

way

for

these

future

explorations.

We

use

the

term

duopolistic

to

denote

the

fact

that

there

are

two

firms

competing

in

the

market,

but

only

for

a

subset

of

the

consumers

who

are

willing

to

buy

more

than

one

good.

Similarly,

we

will

use

the

term

monopolistic

to

denote

the

fact

that

neither

firm

faces

any

competition

in

its

market..

In

our

model,

the

suppliers

choose

the

new

good

price.

In

the

traditional

channel

literature

with

a

decentralized

distribution

structure,

allowing

the

retailer

to

choose

the

price

leads

to

the

well-known

double

marginalization

problem

and

results

in

an

increase

in

new

price

by

an

amount

proportional

to

the

wholesale

price

Because

the

used

prices

are

determined

by

market-clearing

conditions,

this

leads

to

an

increase

in

the

used

price

as

well.

As

a

result,

the

qualitative

nature

of

our

results

remains

unchanged..

This

eventually

results

in

both

suppliers

selling

in

each

market,

which

makes

this

case

less

attractive

for

us

to

analyze..

We

do

not

consider

the

channel

coordination

issue

as

it

is

not

the

focus

of

this

paper.

There

is

a

large

literature

on

channel

coordination

mechanisms

in

marketing

and

operations

for

the

supply

chain

or

sale

channel,

using

tools

such

as

"return

policies"

or

"buy-back

policies."

See,

for

example,

Cachon

A

comparison

of

the

supplier

profits

reveals

that

for

all

α

suppliers

are

better

off

under

a

duopoly

compared

to

a

monopoly.

However,

since

our

main

focus

is

on

how

used

good

markets

affect

supply-chain

profitability,

we

do

not

analyze

this

case

in

detail..

This

is

similar

to

the

approach

taken

by

Desai

and

Purohit

Unlike

their

model,

where

the

manufacturer

chooses

optimal

quantities,

in

our

model,

the

suppliers

choose

optimal

prices..

This

implies

that

below

this

cutoff

value

of

h,

the

suppliers

would

find

it

profitable

not

to

have

the

NH

segment.

We

derive

this

cutoff

value

of

h

in

the

Appendix

and

show

that

it

is

quite

low

for

the

whole

range

of

used

quality

parameter

q..

Note

that

prior

research

by

Rust

has

indicated

that

used

good

markets

will

cause

new

good

prices

to

increase.

In

our

model,

if

consumers

are

unwilling

to

pay

suppliers

for

secondperiod

utility

(i.e.,

if

h

is

equal

to

zero),

then

the

new

good

price

is

p

n

q

-k

q

as

shown

in

the

Appendix.

Hence,

in

our

model

as

well,

opening

a

used

good

market

results

in

the

new

good

price

increasing

from

to

p

n

which

is

consistent

with

prior

findings.

Proof

of

Proposition

has

been

omitted

in

the

Appendix,

because

it

is

immediate.

With

indirect

appropriability,

the

seller

can

extract

all

the

rents

from

a

user

by

charging

a

higher

price

for

the

original

because

the

total

willingness-to-pay

for

the

whole

is

higher..

We

ran

the

same

regression

using

dummy

variables

for

capturing

information

on

book

types-that

is,

we

used

dummies

for

fiction

books,

nonfiction

books,

hardcovers,

paperbacks,

textbooks,

best

sellers,

former

best

sellers,

and

computer

books.

These

serve

as

control

variables

in

addition

to

Datediff.

Again,

there

was

no

qualitative

change

in

the

results.


Uniform convergence of discrete curvatures from nets of curvature lines We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net. The field of discrete differential geometry has brought to light intriguing discrete counterparts of classical differential geometric concepts as well as efficient geometric algorithms (see, e.g., Grinspun and Desbrun, 2006) . One aspect of this theory is convergence: classical smooth notions should arise in the limit of refinement. Recently, several convergence results have been obtained for curvatures and differential operators defined on polyhedral surfaces. Roughly, one may distinguish three approaches: (i) polynomial surface approximation (see, e.g., Meek and Walton, 2000; Cazals and Pouget, 2005) , (ii) geometric measure theory (see, e.g., Fu, 1993; Cohen-Steiner and Morvan, 2006) , and (iii) finite element analysis (see, e.g., Dziuk, 1988; Hildebrandt et al., 2006) . Among these, (i) provides pointwise convergent curvatures for many, but not all, discrete meshes. In contrast, (ii) and (iii) consider generalizations of integrated, or total, curvatures and yield convergence in the sense of measures or appropriate Sobolev norms, respectively.. Given the convergence of curvatures studied by approaches (ii) and (iii) in an integrated sense, it is natural to ask whether these curvatures can be shown to also converge in a pointwise manner. An affirmative answer can be obtained in some special cases, such as polyhedral surfaces with vertices on the unit 2-sphere (Xu, 2006) . In general, however, the answer to this question is negative: it was observed in Xu et al. (2005) that for general irregular polyhedral surfaces, there exist no k-local definitions of discrete curvatures that are pointwise convergent. Here, by k-locality we mean that the definition of curvatures associated with a vertex p of a polyhedral surface only depends on the k-star of p, i.e., those vertices that are connected to p by a path of at most k edges. The concept of k-locality is motivated by the smooth setting, where the definition of curvatures and differential operators only depends on local properties of the underlying Riemannian manifold.. Uniform convergence from nets of curvature lines We provide an affirmative answer to the above question of pointwise convergence of curvatures for a special class of discrete meshes: discrete nets of curvature lines on a given smooth surface M that is immersed into Euclidean space E 3 (see Figure 1) . To obtain approximations (k 1 , k 2 ) of principal curvatures (κ 1 , κ 2 ) on M, we follow a three-step approach. We consider (a) local polyhedral approximations to nets of curvature lines, to which we apply (b) wellknown 1-local integrated notions of discrete curvatures, such as those based on normal cycles (see, e.g., Cohen-Steiner and Morvan, 2006) or those based on the so-called cotangent formula (see, e.g., Pinkall and Polthier, 1993) , followed by (c) dividing the resulting integrated curvatures by appropriate area terms. The resulting pointwise curvature approximations k 1 , k 2 : V → R are at first only defined on the vertex set V of the underlying net. However, we may regard these curvature approximations as functions k i : M → R by extending them from V in a piecewise constant manner to the intrinsic Voronoi regions of the set V on M. Assuming this extension, we show: Theorem 1. Let M be a smooth compact oriented surface without boundary 1 immersed into E 3 . Consider a discrete net of curvature lines on M such that at each vertex the sampling condition (10) is satisfied. Let be an upper bound for the edge lengths of the net such that additionally the intrinsic -balls around vertices cover all of M. Then. where C depends only on properties of M and the shape regularity (9) of the net of curvature lines.. A few remarks seem pertinent before proceeding:. • Uniform pointwise convergence of principal curvatures obtained by a 1-local construction from nets of curvature lines is somewhat surprising since in general 1-local polyhedral curvatures may not even converge in L 2 , even if the mesh vertices reside on the smooth limit surface (see Hildebrandt et al., 2006 ).. • Our pointwise curvature approximations arise from dividing integrated "Steiner-type" curvatures by associated area terms. For example, the integrated mean curvature of an interior edge of a polyhedral surface may be defined as the product between the length of that edge and the signed angle between the normals of its adjacent faces -a definition that arises from Steiner's view of considering offset surfaces. To obtain pointwise curvature approximations from discrete integrated curvatures, we divide by so-called circumcentric areas. While this approach is not new, see, e.g., Desbrun et al. (2005) , our convergence result may be interpreted as a justification of this construction provided that the edges of a polyhedral surface well approximate the principal curvature directions of a smooth limit surface. We prove uniform lower and upper bounds for edge-based circumcentric areas that may be of interest in their own right.. • For our result to hold we require the explicit knowledge of positions of the vertices of a net of curvature lines on a smooth surface as well as the combinatorics of this net. More precisely, our curvature approximations at a vertex p require the position of p and the positions of its direct neighbors (with respect to the combinatorics of the net). Note that we do not require the knowledge of the entire net, lemon star monstar Figure 2 : The three generic patterns of curvature lines near an umbilical point, called lemon, star, and monstar by Berry and Hannay (1977) .. though. (Given such an entire smooth net, it would be trivial to compute the principal curvatures at its vertices.) It would be desirable to drop from our approach the requirement of the exact knowledge of vertex positions of a smooth net of curvature lines. Here, one avenue for further study might be to consider discrete analogues of curvature line nets: so-called principle contact element nets, see Bobenko and Suris (2008) .. Our uniform convergence result given in Theorem 1 is a consequence of a corresponding local error estimate given in Theorem 2. This error estimate holds up to and including umbilical points, where singularities in the curvature line pattern arise. Using a refinement sequence for each of the three surfaces shown in Figure 2 , we observed numerically that although the shape regularity of the net may blow up near umbilics, linear convergence with respect to the maximum edge length remains valid in these cases. Our experiments also indicate that linear convergence is optimal.. Alternative approaches An alternative point of departure for establishing pointwise convergence of discrete curvatures is to give up k-locality and to allow for k → ∞ as the mesh refinement increases. In fact, the above mentioned convergence results of (ii) and (iii) may be interpreted in this way: by decreasing the diameter of the domains over which discrete curvatures are integrated (measured), while simultaneously increasing the mesh refinement inside these domains at a sufficiently fast rate, one recovers classical pointwise notions of smooth curvatures in the limit. In a similar fashion, Belkin et al. (2008) proposed a discrete Laplace operator based on the heat kernel. This operator converges in a pointwise manner if the kernel is scaled down while the mesh resolution is increased sufficiently fast relative to the scaling of the kernel. In contrast to these works, which need to allow for k → ∞ to establish pointwise convergence, our result is obtained by working with the simplest and most local definition: k = 1.

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