Long-term contracts and entry deterrence in the French electricity market - Motivated by recent EU case law, we investigate how long-term contracts may be used as a means of entry deterrence in the French electricity market.

1. Master
APE
September
2014
Long-­‐term
contracts
and
entry
deterrence
in
the
French
electricity
market
Author:
REID,
Christopher
Supervisor:
SPECTOR,
David
Referee:
TROPEANO,
Jean-­‐Philippe
JEL
codes:
D43,
L94
Keywords:
Electricity,
contracts,
market
entry,
simulation

2. 1
Abstract
Motivated
by
recent
EU
case
law,
we
investigate
how
long-­‐term
contracts
may
be
used
as
a
means
of
entry
deterrence
in
the
French
electricity
market.
In
our
model
this
market
consists
of
two
segments:
a
conventional
(e.g.
gas,
coal)
segment
in
which
there
is
perfect
competition,
and
a
nuclear
segment
dominated
by
one
producer.
Our
analysis
is
focused
on
market
entry
in
the
nuclear
segment.
The
nuclear
capacity
that
maximises
the
monopoly
profit
also
minimizes
the
total
cost
of
electricity
production.
Thus,
the
monopoly
capacity
is
efficient.
When
there
is
market
entry,
firms
compete
via
a
discriminatory
auction
mechanism.
We
simulate
the
model,
calibrated
to
the
French
electricity
market.
In
the
absence
of
contracts,
market
entry
leads
to
excess
capacity:
the
total
cost
of
electricity
production
increases,
while
total
profit
and
the
price
of
electricity
decrease.
Long-­‐term
contracts
lead
to
reduced
entry,
but
cannot
eliminate
entry
unless
the
rival
has
large
fixed
costs.
Using
contracts,
the
incumbent
can
increase
its
profit
compared
to
free
entry,
but
cannot
recover
the
monopoly
profit.
The
price
of
electricity
on
the
spot
market
is
not
significantly
affected
by
long-­‐term
contracts.
Overall,
the
welfare
effect
of
long-­‐term
contracts
is
ambiguous.

3. 2
Table
of
Contents
Abstract
..............................................................................................................................................
1
Introduction
.....................................................................................................................................
3
1.
Literature
review
....................................................................................................................
5
2.
Monopoly:
optimal
capacity
................................................................................................
8
2.1.
Electricity
demand
.......................................................................................................................
8
2.2.
Efficient
capacity
level
.............................................................................................................
10
2.3.
Nuclear
capacity
under
monopoly
.......................................................................................
12
3.
Duopoly:
optimal
capacity
and
contracts
....................................................................
14
3.1.
Auction
mechanism
...................................................................................................................
14
3.2.
Large
firm
profit
and
optimal
capacity
...............................................................................
17
3.3.
Small
firm
profit
and
optimal
capacity
...............................................................................
18
3.4.
Long-­‐term
contracts
..................................................................................................................
19
4.
Numerical
simulation
.........................................................................................................
22
4.1.
Calibration
....................................................................................................................................
22
4.2.
Monopoly
......................................................................................................................................
22
4.3.
Duopoly
.........................................................................................................................................
23
4.4.
Long-­‐term
contracts
..................................................................................................................
25
Conclusion
......................................................................................................................................
29
References
.....................................................................................................................................
31
Appendix
–
selected
MATLAB
code
........................................................................................
32

4. 3
Introduction
On
17
March
2010,
the
European
Commission
(EC)
adopted
a
decision1
concerning
the
French
market
for
the
supply
of
electricity
to
large
industrial
customers.
The
Commission
was
concerned
that
EDF
(the
incumbent
operator)
may
have
abused
its
dominant
position
by
concluding
long-­‐term
supply
contracts
which
had
the
effect
of
foreclosing
the
market.
Aaccording
to
the
Commission,
the
volume
and
duration
of
EDF’s
contracts
did
not
provide
sufficient
opportunities
for
alternative
suppliers
to
compete
“for
the
contracts”.
In
addition,
the
exclusionary
nature
of
the
contracts
(whether
through
explicit
provisions
in
the
contract
or
de
facto
exclusivity)
may
have
restricted
competition
“during
the
contracts”.
The
contracts
also
prohibited
clients
from
reselling
electricity,
limiting
customers’
ability
to
manage
their
consumption.
In
response
to
these
objections,
EDF
proposed
a
set
of
commitments,
which
the
Commission
made
legally
binding
in
its
March
2010
decision.
Firstly,
65%
of
the
electricity
supplied
to
large
industrial
customers
would
return
to
the
market
each
year.
Secondly,
the
duration
of
contracts
without
free
opt-­‐out
would
be
limited
to
five
years.
Finally,
EDF
would
allow
competition
during
the
contract
period
by
systematically
proposing
an
alternative
supplier,
enabling
customers
to
source
electricity
simultaneously
from
two
suppliers.
EDF
also
made
a
commitment
to
end
restrictions
on
the
resale
of
electricity
by
clients
under
contract.
This
decision
by
the
EC
forms
the
motivation
for
our
research.
We
investigate
the
impact
of
long-­‐term
supply
contracts
in
a
simple
model
of
the
French
electricity
market.
This
market
is
characterized
by
the
fact
that
nuclear
power
stations
constitute
a
large
proportion
of
installed
generating
capacity.
Although
in
practice
EDF
must
guarantee
competitors
access
to
low-­‐cost
nuclear
electricity,
the
nuclear
capacity
remains
under
the
control
of
EDF,
the
incumbent
operator.
We
consider
here
that
EDF
has
exclusive
rights
over
its
nuclear
capacity,
and
we
focus
our
analysis
on
competition
in
this
segment
of
the
market.
Compared
to
coal
and
gas
power
stations
where
fuel
costs
are
high,
nuclear
power
has
low
operating
costs.
However,
the
investment
costs
for
nuclear
are
much
higher
and
must
be
accounted
for
throughout
the
lifetime
of
the
power
station.
This
feature
of
nuclear
energy
makes
market
entry
particularly
difficult.
We
model
the
French
electricity
market
as
two
segments:
a
conventional
(coal
and
gas)
segment
with
high
operating
costs,
and
a
nuclear
1
see
press
release
IP/10/290
of
17
March
2010
and
Bessot
et
al.
(2010)

5. 4
segment
dominated
by
EDF.
We
consider
that
the
conventional
segment
is
perfectly
competitive:
as
a
result
of
market
entry,
firms
make
zero
profit
in
this
segment.
By
contrast,
the
nuclear
segment
is
a
monopoly:
all
installed
capacity
is
controlled
by
EDF.
In
addition,
EDF
may
sign
long-­‐term
supply
contracts
with
large
industrial
customers.
We
study
market
entry
in
the
nuclear
segment
first
in
the
absence
of
contracts,
and
then
we
introduce
contracts
to
determine
whether
they
may
be
used
to
dissuade
entry.
Our
findings
provide
a
theoretical
basis
for
the
EC’s
decision:
long-­‐term
contracts
do
indeed
have
a
foreclosure
effect
on
the
market,
leading
to
reduced
entry
by
potential
rivals.
In
the
presence
of
large
fixed
costs
or
a
minimum
efficient
scale,
a
sufficient
volume
of
long-­‐term
contracts
may
even
exclude
rivals.
However,
the
welfare
effect
is
ambiguous.
The
capacity
installed
by
the
monopoly
is
optimal
from
the
point
of
view
of
production
efficiency.
Hence,
market
entry
leads
to
excess
capacity,
increasing
the
cost
of
producing
electricity.
However,
market
entry
also
leads
to
lower
total
profits
for
electricity
suppliers
and
reduces
the
average
price
of
electricity.
This
decrease
in
price
may
be
viewed
as
beneficial
from
the
point
of
view
of
consumers,
although
lower
profits
may
lead
to
insufficient
investment
in
the
future
(this
time
dimension
is
absent
from
our
model).
By
decreasing
excess
entry,
long-­‐term
contracts
help
to
minimize
the
total
cost
of
producing
electricity.
Furthermore,
their
effect
on
spot
market
price
is
very
limited
(compared
to
the
spot
market
price
after
market
entry,
in
the
absence
of
contracts).
However,
customers
who
have
signed
a
long-­‐term
contract
are
committed
to
paying
a
high
price
for
electricity
and
cannot
benefit
from
the
reduced
spot
market
price.
The
paper
is
structured
as
follows:
we
begin
by
reviewing
the
literature
on
contracts
as
a
means
of
entry
deterrence.
In
section
two,
we
calculate
the
capacity
that
minimizes
the
total
cost
of
electricity
production,
and
the
capacity
that
maximizes
monopoly
profit.
In
section
three,
we
introduce
a
second
producer
and
derive
analytical
expressions
for
the
two
firms’
profits
and
capacity
choices,
with
and
without
contracts.
In
section
four,
we
simulate
the
model
(calibrated
to
the
French
electricity
market)
and
discuss
the
results.

6. 5
1. Literature
review
Historically,
exclusionary
contracts
have
been
a
contentious
issue
in
antitrust
law
and
scholarship.
Rasmusen,
Ramseyer
and
Wiley
(1991)
cite
several
cases
in
which
US
judges
found
such
contracts
to
be
anticompetitive
and
illegal 1 .
However,
Chicago
School
academics
responded
to
such
cases
with
scepticism.
Director
and
Levi
(1956)
argued
that
customers
would
not
agree
to
sign
exclusionary
contracts
with
a
company
unless
it
offered
them
compensation
for
lost
customer
surplus.
Such
compensation
would
exceed
monopoly
profits,
making
exclusion
too
costly
for
the
incumbent
firm.
In
a
seminal
paper,
Aghion
and
Bolton
(1987)
show
that
exclusionary
contracts
may
in
fact
be
used
profitably
for
entry
deterrence.
In
their
model,
two
buyers
agree
to
sign
an
exclusionary
agreement
despite
jointly
preferring
to
refuse.
The
model
depends
on
three
assumptions:
first,
the
excluding
firm
can
commit
to
a
future
price
level,
and
each
customer
can
escape
the
contract
by
paying
liquidated
damages.
Second,
the
entrant’s
marginal
cost
is
unknown
and
may
be
different
from
the
incumbent
firm’s
marginal
cost,
which
is
constant.
Third,
active
producers
incur
a
fixed
cost,
leading
to
economies
of
scale.
Aghion
and
Bolton
also
allow
the
incumbent
to
make
an
offer
to
one
buyer
that
is
conditional
on
the
other
buyer’s
decision
to
accept
the
offer.
Rasmusen,
Ramseyer,
and
Wiley
(1991
–
we
refer
to
this
paper
as
“RRW”)
show
that
the
incumbent
may
exclude
rivals
by
exploiting
buyers’
lack
of
coordination,
without
requiring
the
previous
assumptions.
Specifically,
if
there
is
a
minimum
efficient
scale2,
the
incumbent
need
only
lock
up
a
proportion
of
the
customers
to
forestall
entry.
“If
each
customer
believes
that
the
others
will
sign,
each
also
believes
that
no
rival
seller
will
enter.
Hence,
a
customer
loses
nothing
by
signing
the
exclusionary
agreement
and
will
indeed
sign.”
Segal
and
Whinston
(2000)
correct
some
errors
in
RRW
and
refine
the
analysis,
focusing
on
how
an
incumbent
can
use
discriminatory
offers
to
exploit
externalities
that
exist
among
buyers.
The
model
has
three
periods,
featuring
three
sets
of
agents:
an
incumbent
firm,
a
potential
rival,
and
a
set
of
buyers.
In
period
one,
the
incumbent
offers
buyers
exclusionary
contracts.
In
period
two,
the
rival
decides
whether
to
enter,
and
in
period
three,
active
firms
compete
à
la
Bertrand.
The
authors
examine
two
different
settings
for
period
one:
1
Examples
include:
U.S.
v.
Aluminum
Co.
of
America
(1945),
Lorain
Journal
Co.
v.
U.S.
(1951),
and
United
Shoe
Machinery
Corp.
v.
U.S.
(1922).
2
RRW
assume
a
minimum
efficient
scale,
but
no
economies
of
scale
beyond
that.
Hence,
exclusion
is
not
simply
the
result
of
a
natural
monopoly.

7. 6
simultaneous
offers
and
sequential
offers.
When
the
incumbent
deals
with
buyers
simultaneously
without
the
ability
to
discriminate,
profitable
exclusion
relies
on
a
lack
of
coordination
among
buyers.
This
is
not
the
case
when
the
incumbent
can
discriminate
between
buyers:
discrimination
allows
the
incumbent
to
exclude
rivals
profitably
by
exploiting
externalities
across
buyers.
When
the
incumbent
deals
with
buyers
sequentially,
its
ability
to
exclude
is
strengthened.
Segal
and
Whinston
show
that
when
the
number
of
buyers
is
large,
the
incumbent
is
able
to
exclude
for
free.
A
related
literature
deals
with
financial
forward
contracts.
Unlike
exclusionary
contracts,
forward
contracts
do
not
forbid
consumers
from
dealing
with
entrants
and
do
not
directly
restrict
a
producer’s
choice
of
output
and
price.
Instead
of
making
legal
restrictions,
they
influence
behaviour
on
the
spot
market
by
altering
incentives
for
firms.
Provided
they
are
observable,
forward
contracts
may
be
used
as
a
signal
of
commitment
to
future
aggressive
behaviour
on
the
spot
market.
In
this
way,
they
may
have
an
entry
deterrence
effect
similar
to
that
of
exclusionary
contracts.
However,
the
effect
is
strongly
dependent
on
whether
firms
compete
in
quantity
(Cournot
competition)
or
price
(Bertrand)
on
the
spot
market.
Allaz
and
Vila
(1993)
show
that
forward
markets
can
improve
the
efficiency
of
production
decisions
in
a
Cournot
duopoly.
They
begin
by
noting
that
“usually
appearance
of
forward
markets
is
justified
by
agents’
desire
to
hedge
risk”,
requiring
uncertainty
over
some
variable.
Allaz
and
Vila
show
that
this
is
not
necessary:
forward
markets
can
be
used
under
certainty
and
perfect
foresight.
Producers
use
forward
transactions
as
strategic
variables.
The
authors
first
consider
a
two-­‐period
model
of
duopoly
with
linear
costs
and
demand,
under
perfect
foresight
and
certainty.
Firms
choose
forward
positions
in
period
one
and
produce
in
period
two.
The
firm
with
access
to
the
forward
market
gains
first-­‐mover
advantage
(becomes
Stackelberg
leader
on
the
spot
market).
However,
a
prisoner’s
dilemma
arises
when
both
firms
have
access
to
the
forward
market:
a
firm
greatly
benefits
from
being
the
only
producer
to
trade
forward,
but
if
both
firms
trade
forward,
they
end
up
worse
off.
The
authors
then
extend
the
model
to
trading
periods,
and
show
that
when
tends
to
infinity,
the
competitive
outcome
is
obtained.
Mahenc
and
Salanié
(2004)
investigate
a
model
in
which
duopolists
producing
two
differentiated
goods
can
trade
forward
before
competing
à
la
Bertrand
on
spot
markets.
Similarly
to
Allaz-­‐Vila,
the
model
features
two
periods:
in
period
one
each
firm
takes
a
position
on
the
forward
market,
and
in
period
two
they
compete
on
the
spot
market.
However,
the
crucial
difference
is
that
competition
on
the
spot
market
is
à
la
Bertrand:
firms
choose
prices,
not
quantities,
and
the
goods
are
not
perfect
substitutes.
Mahenc
and
Salanié
reach
a
conclusion
that
is
opposite
to
that
of
Allaz
and
Vila:
in
equilibrium
firms
buy
forward
their
own
production,
leading
to
higher
spot
prices
than
in
the
static
case
(no
forward
market).
Hence,
forward

8. 7
markets
have
a
softening
effect
on
competition
in
this
case.
This
competition-­‐softening
effect
is
stronger
when
competition
increases,
that
is
when
goods
are
more
substitutable.
Lien
(2000)
analyses
the
role
of
forward
contracts
in
the
electricity
market.
He
argues
that
there
is
a
“curse
of
market
power”:
in
the
short
term,
large
firms
have
an
incentive
to
hold
back
output
in
order
to
push
up
prices.
However,
this
leads
to
excess
entry
by
small
producers,
who
benefit
from
high
prices
without
incurring
the
costs
of
restricted
output.
As
a
result,
long-­‐
term
profits
of
large
firms
are
reduced.
Lien
shows
that
forward
sales
can
eliminate
this
curse
by
deterring
excess
entry.
Our
analysis
differs
from
that
of
Lien
in
two
important
ways:
firstly,
we
concentrate
on
the
French
electricity
market
and
in
particular
the
nuclear
segment
of
this
market.
The
French
market
is
characterised
by
the
high
proportion
of
electricity
that
is
provided
by
nuclear
power
stations.
Given
the
high
investment
costs
associated
with
nuclear
energy,
entry
is
particularly
difficult
in
this
segment
of
the
market.
Nuclear
power
stations
typically
provide
baseload
power
and
are
operational
most
of
the
time,
which
makes
the
use
of
long-­‐term
supply
contracts
particularly
convenient.
Secondly,
we
model
competition
in
the
electricity
market
using
an
auction
mechanism
studied
by
Fabra,
von
der
Fehr,
and
Harbord
(2006).
There
are
two
producers
in
this
model.
Each
producer
submits
a
bid
(offer
price)
to
a
central
auctioneer,
who
then
allocates
production
in
order
to
meet
demand.
Fabra
et
al.
study
two
mechanisms:
a
uniform
auction,
in
which
all
active
producers
(those
whose
bid
is
wholly
or
partly
accepted)
are
paid
the
same
price,
and
a
discriminatory
auction,
in
which
active
producers
are
paid
their
offer
price.
The
authors
find
that
uniform
auctions
result
in
higher
prices
than
discriminatory
auctions.
In
a
related
paper,
Fabra,
von
der
Fehr,
and
de
Frutos
(2011)
study
the
impact
of
the
auction
format
on
investment
incentives.
They
find
that
investment
incentives
are
(weakly)
stronger
under
discriminatory
auctions
than
under
uniform
auctions.
For
this
reason,
we
focus
on
discriminatory
auctions.

9. 8
2. Monopoly:
optimal
capacity
In
our
model,
the
French
electricity
market
consists
of
two
segments:
a
perfectly
competitive
conventional
segment,
and
a
nuclear
segment
dominated
by
one
firm.
Marginal
costs
are
constant:
conventional
electricity
costs
to
produce,
and
nuclear
electricity
costs
.
Operating
costs
for
conventional
generation
are
higher
than
for
nuclear
generation,
so
.
The
conventional
sector
is
perfectly
competitive,
so
firms
make
zero
profit.
However,
the
nuclear
sector
is
run
by
a
monopoly.
Before
considering
entry
by
potential
rivals
in
the
nuclear
sector,
we
determine
the
optimal
nuclear
capacity
from
the
point
of
view
of
social
welfare
and
from
the
monopoly’s
point
of
view.
2.1.
Electricity
demand
In
order
to
calculate
optimal
capacity,
we
need
to
characterise
electricity
demand.
Compared
to
demand
in
markets
for
goods,
electricity
demand
is
unusual.
Firstly,
demand
is
not
in
terms
of
quantity
but
in
terms
of
rate.
Indeed,
quantity
is
expressed
in
terms
of
energy,
with
units
of
GWh
(gigawatt-­‐hour)
for
example,
whereas
demand
and
supply
are
expressed
in
terms
of
power,
with
units
of
GW
(gigawatt).
This
is
because
of
a
physical
property
of
the
system:
unlike
goods,
electricity
cannot
be
stored.
Hence,
the
rate
at
which
electricity
is
provided
to
the
network
must
equal
the
rate
at
which
it
is
consumed
by
customers.
Secondly,
it
is
almost
perfectly
inelastic.
That
is,
demand
does
not
change
in
response
to
price.
The
costs
and
given
above
are
expressed
in
monetary
units
per
quantity
of
electricity,
for
example
million
euros
per
GWh
(€m/GWh).
However,
this
is
equivalent
to
paying
for
capacity
for
a
certain
time:
if
a
firm
supplies
10
GW
for
2
hours
at
a
cost
of
0.03
€m/GWh,
it
will
incur
a
cost
of
€
600,000.
We
use
data
on
electricity
consumption
in
2012
obtained
from
RTE,
the
electricity
transmission
system
operator
of
France1.
The
data
describes
electricity
consumption
in
MW
for
every
half
hour
period
of
the
year
(the
data
is
described
in
greater
depth
in
Appendix
A).
Figure
1
shows
the
distribution
of
electricity
demand
as
a
histogram.
1
Source:
http://clients.rte-­‐france.com/lang/fr/clients_producteurs/vie/vie_stats_conso_inst.jsp,
last
accessed
on
21/08/2014.

10. 9
30 40 50 60 70 80 90 100 110
0
10
20
30
40
50
60
70
Electricity demand (GW)
Frequency(days)
Figure
1
–
Distribution
of
electricity
demand
in
2012:
histogram
In
order
to
proceed
with
the
analysis,
we
need
to
fit
a
known
distribution
function
to
the
data.
It
is
important
to
note
that
electricity
demand
is
not
random;
indeed,
it
can
be
accurately
forecast.
However,
we
use
a
probability
distribution
function
(pdf)
in
our
analysis
because
we
want
to
calculate
quantities
such
as
profit
analytically.
In
what
follows,
we
normalize
everything
by
time:
quantities
are
expressed
per
year,
and
we
consider
2012
as
a
typical
year.
For
convenience,
we
use
a
uniform
probability
distribution.
Figure
2
shows
a
kernel
estimate
of
the
pdf
as
well
as
the
“best
fit”
uniform
pdf.
The
parameters
of
the
uniform
pdf
are
and
.
We
choose
these
in
order
to
match
the
mean
and
standard
deviation
of
observed
demand,
as
given
in
table
1
(rounded
to
the
nearest
GW).
Name
Value
(GW)
33
78
Mean
55.5
Standard
deviation
13
Table
1
-­‐
Electricity
demand
parameters

11. 10
Figure
2
–
Distribution
of
electricity
demand
in
2012:
kernel
density
and
“best
fit”
uniform
pdf
2.2. Efficient
capacity
level
We
begin
by
determining
the
level
of
nuclear
capacity
that
is
optimal
from
the
point
of
view
of
social
welfare.
We
call
this
the
efficient
capacity
level,
and
it
maximizes
total
surplus:
Roughly
speaking,
consumer
surplus
is
the
difference
between
indirect
utility
derived
from
consumption
of
electricity
and
the
cost
of
purchasing
electricity.
Usually,
we
would
have
to
integrate
over
price
or
capacity
(since
capacity
has
an
impact
on
price).
However,
demand
for
electricity
is
perfectly
inelastic,
and
we
assume
that
conventional
producers
have
an
infinite
capacity,
meaning
that
it
is
large
enough
to
supply
any
level
of
consumer
demand.
Hence,
there
is
no
need
for
demand
rationing.
As
a
result,
the
indirect
utility
does
not
vary
with
price
or
nuclear
capacity.
Producer
surplus
is
equal
to
the
sum
of
profits
of
electricity
producers
(conventional
and
nuclear):
the
difference
between
total
revenue
and
the
total
cost
of
producing
electricity.
Total
revenue
of
firms
is
equal
to
total
expenditure
by
consumers,
so
we
have:

12. 11
As
noted
above,
the
indirect
utility
is
constant,
so
maximizing
total
surplus
is
equivalent
to
minimizing
the
total
cost
of
producing
electricity.
We
denote
nuclear
capacity
by
,
and
electricity
demand
by
.
When
demand
is
less
than
nuclear
capacity
( ,
then
demand
is
met
entirely
by
electricity
from
nuclear
power
stations,
which
have
operating
cost
.
When
demand
exceeds
nuclear
capacity
( ),
conventional
power
stations
are
required
to
supply
the
excess
( ),
with
an
operating
cost
of
.
In
addition,
nuclear
capacity
has
a
yearly
investment
cost
of
per
GW.
Nuclear
power
stations
require
significant
capital
to
build.
This
capital
usually
takes
the
form
of
a
loan,
which
is
reimbursed
in
instalments
during
the
lifetime
of
the
power
station.
The
investment
cost
represents
capital
cost
repayments,
and
also
includes
yearly
maintenance
and
fuel
costs.
Capital
costs
for
conventional
power
stations
are
less
significant
than
for
nuclear,
whereas
operating
costs
are
higher.
For
this
reason,
we
ignore
investment
costs
for
conventional
power.
The
total
cost
of
producing
electricity
includes
operating
costs
for
nuclear
and
conventional
power,
and
investment
costs
for
nuclear
power.
We
use
the
uniform
distribution
for
electricity
demand
described
above
to
calculate
the
total
cost
over
the
year,
which
we
denote
by
:
Where
denotes
the
pdf
of
electricity
demand,
setting
:
Evaluating
the
integrals
and
setting
,
we
have:
The
efficient
capacity,
,
minimizes
this
cost.
Differentiating
with
respect
to
,
we
have:
Setting
,
we
find
the
efficient
nuclear
capacity
level:

13. 12
The
efficient
capacity
depends
on
the
ratio
.
The
denominator
is
the
difference
between
conventional
and
nuclear
operating
costs;
as
we
will
see
in
the
next
section,
it
is
also
the
difference
between
the
electricity
price
under
monopoly
(in
the
nuclear
sector)
and
the
nuclear
operating
cost.
If
,
then
and
:
in
the
absence
of
investment
costs,
the
least
costly
option
is
for
all
electricity
to
be
produced
from
nuclear
energy.
On
the
other
hand,
if
,
then
and
:
nuclear
capacity
is
always
in
use
providing
baseload
power,
while
conventional
capacity
is
used
to
supply
,
the
variable
part
of
demand.
2.3. Nuclear
capacity
under
monopoly
We
now
calculate
the
capacity
that
maximizes
the
profit
of
a
nuclear
producer
with
a
monopoly.
The
conventional
sector
remains
perfectly
competitive,
but
there
is
a
single
producer
of
nuclear
electricity.
The
profit
of
a
nuclear
monopolist,
which
we
denote
,
is
given
by
the
following
expression,
taking
into
account
both
operating
costs
and
investment
costs:
When
,
nuclear
capacity
can
cover
consumer
demand
entirely.
When
demand
exceeds
nuclear
capacity
( ),
nuclear
capacity
is
saturated.
Since
the
conventional
sector
is
perfectly
competitive,
the
price
of
electricity
produced
by
conventional
means
equals
its
marginal
cost,
.
Hence,
the
nuclear
monopolist
can
charge
up
to
for
its
electricity,
and
to
maximize
profit,
it
will
charge
exactly
.
In
theory,
consumers
would
be
indifferent
between
purchasing
nuclear
and
purchasing
conventional
electricity
if
their
prices
are
equal.
However,
the
network
operator
prioritizes
electricity
produced
with
least
marginal
cost,
so
conventional
power
stations
produce
only
when
nuclear
capacity
is
saturated.
Evaluating
the
integrals,
we
find
the
following
expression
for
monopoly
profit:
Differentiating,
we
have:

14. 13
Setting
the
derivative
to
zero,
we
find
the
monopoly
profit-­‐maximizing
nuclear
capacity:
We
notice
that
the
nuclear
capacity
chosen
by
a
profit-­‐maximizing
monopolist
is
equal
to
the
efficient
generating
capacity.
In
other
words,
the
monopoly
chooses
a
nuclear
capacity
that
minimizes
the
total
cost
of
producing
electricity
(including
both
nuclear
and
conventional).
The
profits
earned
by
the
nuclear
monopolist
and
the
total
cost
of
producing
electricity
are
related
by:
The
sum
of
monopoly
profit
and
the
total
cost
of
producing
electricity
is
,
a
constant.
Hence,
maximizing
is
equivalent
to
minimizing
.
is
the
total
payment
from
electricity
consumers
to
electricity
producers.
Since
the
nuclear
producer
charges
,
the
price
of
power
is
whether
its
source
is
nuclear
or
conventional.
The
payment
covers
the
total
cost
of
production
plus
a
monopoly
rent
for
the
nuclear
generator
(conventional
generators
make
zero
profit).
We
evaluate
monopoly
profit
when
,
and
let
.
We
call
this
“optimal
profit”.
Normalizing
by
,
we
have:
Setting
,
one
can
see
that
maximal
profit
is
a
quadratic
function
of
the
ratio
,
which
we
call
“cost
ratio”
and
denote
by
.
Figure
4
displays
this
function.
As
discussed
previously,
optimal
capacity
is
a
linear
decreasing
function
of
.
When
investment
costs
are
nil
( ,
then
,
and
optimal
profit
is
maximized
(optimal
cost
is
minimized).
is
the
cost
of
adding
a
marginal
unit
of
capacity,
and
is
the
maximum
profit
that
may
be
derived
from
it
(i.e.
if
the
unit
operates
permanently).
Optimal
profit
is
zero
when
these
are
equal

15. 14
( ),
and
.
Optimal
nuclear
capacity
is
zero
( )
when
.
Then
optimal
cost
is
maximal1.
3. Duopoly:
optimal
capacity
and
contracts
3.1.
Auction
mechanism
We
now
introduce
two
firms
in
the
nuclear
sector.
Unlike
typical
markets
(e.g.
for
goods),
they
do
not
compete
directly,
whether
by
price
or
by
quantity.
Instead,
they
compete
via
a
centralized
auction
mechanism.
Fabra,
von
der
Fehr,
and
Harbord
(2006)
study
two
auction
mechanisms
for
the
electricity
market:
uniform
and
discriminatory
auctions.
The
duopoly
comprises
a
large
supplier,
with
capacity
,
and
a
small
supplier,
with
capacity
( ).
We
assume
capacity
is
perfectly
divisible.
The
two
suppliers
compete
by
submitting
bids,
or
offer
prices,
to
the
auctioneer.
The
suppliers
incur
the
same
marginal
cost
for
production
below
capacity,
and
cannot
produce
above
capacity.
We
denote
their
bids
by
.
As
before,
there
is
a
perfectly
competitive
conventional
sector
with
marginal
cost
,
so
prices
in
the
nuclear
sector
cannot
exceed
.
The
level
of
demand
is
,
and
total
nuclear
1
Optimal
profit
is
negative
when
and
jumps
to
zero
when
( .
Figure
3
-­‐
Optimal
profit
as
a
function
of
the
cost
ratio

16. 15
capacity
is
.
We
let
.
The
auctioneer
allocates
between
the
two
nuclear
producers.
If
demand
exceeds
the
total
nuclear
capacity
( )
then
,
and
the
excess
( )
is
dispatched
to
conventional
power
stations.
Output
allocated
to
nuclear
supplier
,
,
is
denoted
by
.
It
is
determined
as
follows:
If
firms
submit
different
bids,
the
lower-­‐bidding
firm’s
capacity
is
dispatched
first.
If
demand
is
in
excess
of
this
capacity,
then
the
higher-­‐bidding
firm
serves
residual
demand.
If
both
firms
submit
the
same
bid,
then
demand
is
split
between
them.
Fabra
et
al.
(2006)
study
two
types
of
auction
mechanisms,
which
differ
in
the
payments
received
by
firms
but
not
in
the
quantities
dispatched:
in
a
uniform
auction,
the
price
received
by
an
active
supplier
is
equal
to
the
highest
accepted
bid
in
the
auction.
In
a
discriminatory
auction,
the
price
received
by
an
active
supplier
is
equal
to
its
own
offer
price,
so
supplier
’s
profit
is
given
by
.
The
equilibrium
outcomes
of
the
auction
are
summarized
in
Proposition
3
of
Fabra,
von
der
Fehr,
and
de
Frutos
(2011)1.
The
authors
distinguish
three
regions
of
demand:
• Low
demand:
.
In
this
region,
either
producer
is
able
to
supply
the
market
fully.
In
other
words,
there
is
no
residual
demand.
The
result
is
equivalent
to
Bertrand
competition
with
perfectly
substitutable
goods
(indeed,
electricity
produced
by
supplier
is
indistinguishable
from
that
produced
by
).
The
suppliers
undercut
each
other
until
they
reach
their
marginal
cost
of
production,
.
In
equilibrium,
both
suppliers
place
bids
at
.
They
produce
a
quantity
each
and
earn
zero
profits.
• High
demand:
.
In
this
region,
at
least
one
of
the
suppliers
is
unable
to
supply
the
market
fully.
The
authors
distinguish
two
regions
within
high
demand:
when
(region
I),
producer
1
can
supply
the
market
fully,
but
producer
2
cannot.
When
(region
II),
neither
producer’s
capacity
is
sufficient
to
cover
demand
entirely,
so
there
is
always
residual
demand
for
the
other.
When
demand
is
high
( ),
there
is
no
pure-­‐
strategy
equilibrium.
Instead
there
is
a
unique
mixed-­‐strategy
equilibrium,
in
which
the
two
1
Proofs
and
equilibrium
strategies
are
given
in
Fabra,
von
der
Fehr,
and
Harbord
(2006)

17. 16
firms
mix
over
a
common
support
that
lies
above
marginal
costs
and
includes
.
The
firms
mix
according
to
different
probability
distributions:
in
particular,
the
large
firm
has
a
mass
point
at
,
the
upper
bound.
The
small
firm
bids
below
with
probability
1,
so
profits
of
the
large
firm
are
the
same
as
if
it
offered
to
sell
residual
demand
at
.
• Very
high
demand:
.
Nuclear
capacity
is
insufficient
to
supply
the
market,
so
conventional
producers
must
supply
residual
demand.
In
equilibrium,
both
nuclear
firms
place
bids
at
and
produce
at
full
capacity.
Intuitively,
it
is
easy
to
understand
why
there
is
no
pure-­‐strategy
equilibrium
in
the
high
demand
region.
Consider
an
initial
situation
where
both
firms
bid
.
Then
either
of
the
suppliers
can
increase
its
profit
by
placing
a
bid
just
below
:
the
increase
in
output
outweighs
the
decrease
in
price.
Let
firm
place
a
bid
just
below
.
Then
the
other
supplier
(firm
),
serving
residual
demand
(which
may
be
zero),
would
benefit
by
placing
a
bid
just
below
that
of
firm
.
The
firms
place
subsequently
lower
bids,
until
the
large
firm
would
profit
more
from
serving
residual
demand
at
than
undercutting
the
small
firm.
But
if
the
large
firm
places
a
bid
at
,
the
small
firm
will
place
a
bid
just
below
,
and
so
on.
The
equilibrium
profits
are
summarized
in
table
2.
We
denote
firm
’s
instantaneous
profit
by
.
This
is
the
profit
obtained
for
a
given
realisation
of
demand,
per
unit
time,
not
including
investment
costs.
Both
firms’
profit
functions
are
continuous
and
increasing
in
.
The
large
firm’s
profit
is
linear
in
and
goes
from
zero
(when
)
to
,
when
.
The
small
firm’s
profit
is
always
less
than
firm
1
profit.
When
demand
is
high,
firm
2
is
concave
hyperbolic
(region
I)
then
linear
(region
II).
Region
Demand
Profits
Low
demand
High
demand
I
High
demand
II
Very
high
demand
Table
2
-­‐
Instantaneous
profits
as
a
function
of
demand

18. 17
3.2. Large
firm
profit
and
optimal
capacity
Having
described
the
auction
mechanism
and
instantaneous
profits,
we
turn
our
attention
to
each
firm’s
total
profit
and
optimal
capacity
choice
under
duopoly.
Both
firms
have
constant
marginal
costs
of
investment
with
a
value
of
.
• If
,
firm
1’s
profit
over
the
year
is
given
by:
• If
,
the
expression
for
firm
1
profit
is
different:
The
expression
is
different
in
that
the
lower
bound
of
the
first
integral
is
instead
of
.
This
arises
because
firm
1
profit
is
zero
when
demand
is
in
the
low
region
( ).
When
,
demand
is
never
in
this
region
(we
always
have
).
To
summarize,
firm
1
profit
is
given
by
the
following
function:
If
we
fix
,
one
can
see
that
is
a
continuous
function
of
that
is
quadratic
when
and
linear
when
.
In
order
to
determine
firm
1’s
optimal
choice
of
capacity,
we
differentiate
with
respect
to
:

19. 18
Capacity
is
optimal
for
firm
1
when
its
marginal
benefit
equals
its
marginal
cost
.
We
assume
,
which
ensures
that
firm
1
makes
positive
profit
when
it
has
a
monopoly.
Setting
,
we
find:
We
have
.
In
other
words,
when
we
recover
the
monopoly
capacity,
which
we
denote
by
.
Interestingly,
it
is
optimal
for
firm
1
to
keep
aggregate
capacity
at
the
efficient
level,
.
If
we
assume
that
firm
1
has
the
monopoly
capacity
(as
we
will
do
when
we
introduce
contracts),
any
entry
by
firm
2
would
lead
to
excess
capacity,
which
is
suboptimal
for
firm1.
So
firm
1
would
prefer
to
give
some
of
its
capacity
to
firm
2
(along
with
the
associated
investment
costs)
rather
than
suffer
the
costs
of
excess
capacity.
Indeed,
we
find
that
.
Evaluating
this
expression
when
and
,
we
find:
Since
,
both
expressions
are
negative:
firm
1
profit
decreases
whenever
there
is
entry
by
firm
2.
However,
firm
1
profit
decreases
faster
when
entry
leads
to
excess
capacity:
when
,
firm
1
gives
capacity
to
firm
2.
Total
capacity
is
constant,
and
firm
1
profit
decreases
linearly.
In
contrast,
when
,
total
capacity
increases
when
firm
2
enters,
and
firm
1
profit
decreases
quadratically.
3.3. Small
firm
profit
and
optimal
capacity
In
order
to
analyse
market
entry
by
the
small
firm,
we
calculate
its
profit
function
and
optimal
capacity
choice.
As
before,
we
distinguish
two
cases.
• If
,
firm
2’s
profit
over
the
year
is
given
by:

20. 19
If we fix , firm 2 profit is a cubic function of . Differentiating with respect to , we find:
Setting
,
firm
2’s
optimal
capacity
choice
is
the
solution
to
the
following
quadratic
equation:
• If
,
the
expression
for
firm
2
profit
is:
As
before,
the
lower
bound
of
the
first
integral
is
instead
of
.
Evaluating
this
expression,
we
find:
This
expression
is
similar
to
the
one
found
previously,
but
the
term
multiplying
is
instead
of
.
More
importantly,
the
term
multiplying
is
now
a
logarithmic
function
of
:
.
This
makes
it
impossible
to
solve
analytically
for
such
that
.
This
will
have
to
be
done
numerically.
As
for
firm
1,
firm
2’s
profit
is
a
continuous
function
of
and
,
which
we
denote
by:
3.4. Long-­‐term
contracts
We
now
introduce
long-­‐term
contracts
to
the
model.
We
assume
that
firm
1
has
had
a
monopoly
in
the
nuclear
sector
for
a
long
time.
Hence,
it
has
had
time
to
build
capacity
up
to
a

21. 20
level
that
maximises
its
profit.
Hence,
we
let
from
now
on.
The
timing
of
the
model
is
as
follows:
1. Firm
1
has
a
monopoly
and
chooses
a
volume
of
long-­‐term
contracts.
2. Firm
2
observes
these
contracts,
and
chooses
how
much
capacity
to
build.
3. The
two
firms
compete
on
the
spot
market
using
the
discriminatory
auction
mechanism
described
previously.
The
contracts
are
“long
term”
in
the
sense
that
they
are
still
in
effect
at
the
time
of
entry.
The
contracts
stipulate
that
firm
1
supplies
a
constant
level
of
power
to
customers
throughout
the
year
at
a
price
.
The
total
capacity
supplied
to
customers
under
contract
is
(we
call
this
the
“volume
of
contracts”).
Hence,
firm
1
has
a
capacity
available
to
compete
on
the
market.
As
a
result,
total
nuclear
capacity
on
the
spot
market
is
reduced
to
.
As
before,
demand
is
uniformly
distributed
between
and
.
This
is
equivalent
to
saying
that
demand
is
the
sum
of
two
components:
a
constant
component
and
a
variable
part
( )
uniformly
distributed
between
and
.
The
constant
component
represents
baseload
power:
for
example,
industrial
consumers
who
use
electricity
at
a
constant
rate
throughout
the
year.
Long-­‐term
supply
contracts
are
signed
between
such
industrial
consumers
and
firm
1.
This
removes
a
volume
of
capacity
from
the
spot
market,
so
spot
market
demand
is
now
distributed
between
and
.
We
place
the
following
restrictions
on
the
volume
of
contracts:
must
be
non-­‐negative
( )
and
cannot
exceed
baseload
power
( ).
This
ensures,
respectively,
that
firm
1
always
supplies
electricity
to
contract
customers
(never
the
other
way
round),
and
that
spot
market
demand
is
always
positive.
Firm
1’s
profit,
taking
into
account
long-­‐term
contracts
and
investment
costs,
is
given
by:
The
function
represents
operating
profits
from
spot
market
competition.
Its
expression
is
the
same
as
the
expression
for
given
previously,
except
that
,
,
and
are
replaced
with
,
,
and
.
Hence,
when
,
we
have:

22. 21
Developing
this
expression,
we
find:
We
note
that
when
.
One
can
see
that
if
firm
1
anticipates
that
will
be
small
( ),
then
firm
1’s
motive
to
sell
supply
contracts
is
purely
strategic.
Indeed,
if
we
ignore
the
impact
of
on
firm
2’s
choice
of
capacity
(taking
as
constant),
then
firm
1
cannot
increase
its
profit
by
selling
contracts.
In
fact,
its
profit
will
be
reduced
if
.
However,
firm
1
may
have
an
incentive
to
sell
contracts
if
it
reduces
entry
by
firm
2
–
this
is
what
we
seek
to
find
out.
Similarly
to
firm
1,
firm
2’s
profit
function,
including
contracts
and
investment
costs,
is:
The
expression
of
is
found
by
replacing
,
,
and
with
,
,
and
in
the
expression
of
.
We
define
firm
2’s
optimal
capacity
choice,
taking
into
account
contracts,
as
follows:
Finally,
we
define
and
:

23. 22
4. Numerical
simulation
4.1. Calibration
We
calibrate
the
model
using
data
for
the
French
electricity
market,
then
simulate
using
MATLAB.
We
have
already
determined
and
,
the
parameters
of
the
electricity
demand
distribution.
The
total
nuclear
capacity
installed
in
France
is
63,130
MW
(source:
RTE1).
We
assume
that
this
capacity
was
chosen
optimally
by
the
monopoly:
GW
We
set
,
using
the
investment
cost
as
a
numéraire,
and
solve
the
previous
equation
to
find
.
These
numbers
are
summarized
in
table
3.
Name
Value
33
GW
78
GW
63
GW
3
1
(numéraire)
Table
3
–
Parameters
of
the
calibrated
model
4.2. Monopoly
The
monopoly
profit
(after
investment
costs)
is
displayed
in
figure
4
as
a
function
of
nuclear
capacity.
It
is
at
a
maximum
when
the
monopoly
has
a
capacity
of
63
GW.
Interestingly,
monopoly
is
negative
when
capacity
is
less
than
9.5
GW.
This
suggests
that
there
is
a
minimum
efficient
scale
for
nuclear
power.
When
GW,
the
value
of
monopoly
profit
is
GW.
As
discussed
in
the
previous
section,
the
monopoly
capacity
is
efficient
in
that
it
minimizes
the
total
cost
of
producing
electricity.
1
http://clients.rte-­‐france.com/lang/an/clients_producteurs/vie/prod/parc_reference.jsp,
last
accessed
on
21/08/2014

24. 23
4.3.Duopoly
Firm
1,
the
ex-­‐monopoly,
has
capacity
GW.
When
firm
2
enters
the
market,
the
two
producers
compete
via
the
discriminatory
auction
mechanism
described
in
the
previous
section.
Figure
5
displays
both
firms’
profit
as
a
function
of
,
firm
2’s
capacity.
Figure
5
–
Firm
1
and
firm
2
profit
(after
investment
cost)
when
,
as
a
function
of
Figure
4
-­‐
Monopoly
profit
(after
investment
costs)
as
a
function
of
nuclear
capacity

25. 24
Firm
1
profit
is
strictly
decreasing
in
,
and
becomes
negative
when
GW.
In
the
absence
of
contracts,
firm
2
profit
is
maximum
when
GW,
so
we
have
GW.
At
this
point,
firm
1
makes
a
profit
of
51,
about
half
of
monopoly
profit.
We
notice
that
firm
2
makes
non-­‐negative
profit
as
long
as
,
which
implies
that
there
is
no
minimum
efficient
scale
for
firm
2.
This
is
because
we
have
not
given
firm
2
any
fixed
costs
–
investment
costs
are
proportional
to
capacity.
However,
if
firm
2
had
fixed
costs
of
say
10,
then
capacity
below
5
GW
would
not
be
profitable.
Figure
6
shows
the
total
profit
of
nuclear
firms
and
the
total
cost
of
electricity
production.
It
also
displays
total
revenue
earned
by
both
nuclear
and
conventional
power
producers,
given
by
the
following
expression:
We
denote
total
revenue
.
As
conventional
power
producers
make
zero
profit,
we
have:
Figure
6
–
Total
profit,
cost,
and
revenue
for
electricity
producers
(both
nuclear
and
conventional)
when
,
as
a
function
of
firm
2
capacity
( )
Total
profit
begins
at
96
(monopoly
profit)
and
decreases
with
.
When
GW,
total
profit
is
70.
In
layman
terms,
the
two
producers
must
share
a
pie
that
decreases
in
size
as
firm
2
enters
the
market.
Total
cost
increases
with
.
Since
the
monopoly
capacity
also
minimises
total
cost,
entry
by
firm
2
leads
to
excess
capacity
and
higher
total
cost.

26. 25
Total
revenue
decreases
with
,
which
implies
that
the
decrease
in
total
profit
is
not
only
associated
with
increased
cost
of
electricity
production.
There
is
also
a
price
effect.
We
define
a
price
index
by
the
following
expression:
.
This
index
of
the
wholesale
price
of
electricity
is
proportional
to
total
revenue.
when
,
and
when
GW.
Hence,
market
entry
by
firm
2
leads
to
higher
total
cost
of
electricity,
and
lower
total
profit
and
revenue.
The
price
of
electricity
decreases
by
approximately
10%.
4.4. Long-­‐term
contracts
We
now
allow
firm
1
to
hold
a
volume
of
long-­‐term
supply
contracts,
according
to
which
firm
1
supplies
electricity
at
a
price
.
At
the
time
of
signing
the
contracts,
firm
1
has
a
monopoly
and
the
price
of
electricity
is
,
so
customers
are
indifferent
between
purchasing
electricity
on
the
market
and
a
contract
where
.
We
set
and
calculate
,
firm
2’s
optimal
choice
of
capacity
as
a
function
of
the
volume
of
contracts
held
by
firm
1.
In
order
to
do
so,
we
use
the
following
program
for
every
value
of
:
1. We
calculate
)
for
a
range
of
values
of
taken
in
the
interval
.
2. We
find
,
the
value
of
corresponding
to
the
maximum
.
3. If
,
we
return
the
analytical
solution.
If
not,
we
return
the
numerical
solution,
.
Figure
7
–
Capacity
chosen
by
firm
2
as
a
function
of
the
volume
of
contracts
held
by
firm
1

27. 26
Figure
7
displays
for
.
The
capacity
chosen
by
firm
2
is
strictly
decreasing
in
.
There
is
a
change
in
slope
when
GW.
Beyond
this
point,
.
As
discussed
in
the
previous
section,
the
profit
functions
of
the
firms
change
when
.
As
a
result,
the
slope
of
changes.
We
note
that
the
reduction
in
firm
2’s
capacity
is
approximately
proportional
to
,
the
volume
of
contracts
expressed
as
a
proportion
of
firm
1
capacity.
Indeed,
when
,
firm
2’s
capacity
is
reduced
by
58%.
Figure
8
–
Profit
of
each
firm
as
a
function
of
the
volume
of
contracts
held
by
firm
1
Figure
8
displays
and
.
Firm
2
profit
is
decreasing
in
,
but
it
remains
positive,
so
although
contracts
decrease
entry,
firm
1
cannot
exclude
firm
2
completely
using
contracts.
However,
if
firm
2
had
large
fixed
costs,
total
exclusion
would
be
possible.
For
example,
if
firm
2
had
fixed
costs
of
10,
then
it
would
not
enter
the
market
if
GW.
Firm
1
profit,
including
income
from
contracts,
is
increasing
in
,
but
remains
less
than
monopoly
profit.
In
order
to
maximize
its
profit,
firm
1
should
choose
a
volume
of
contracts
GW.
At
this
point,
firm
1
makes
a
profit
of
82,
just
14
less
than
monopoly
profit.
Figure
9
displays
total
profit
and
total
cost
as
a
function
of
.
Because
contracts
lead
to
decreased
entry
by
firm
2
(hence,
less
excess
capacity),
the
total
cost
of
electricity
production
decreases
with
.
Total
profit
increases
with
:
the
increase
in
firm
1
profit
outweighs
the
decrease
in
firm
2
profit.

28. 27
Figure
10
displays
total
revenue,
with
and
without
income
from
contracts.
The
difference
is
striking.
Total
revenue,
excluding
contracts,
is
sharply
decreasing.
This
represents
revenue
from
the
spot
market,
whose
size
is
being
reduced
as
the
volume
of
contracts
increases
(peak
demand
in
the
spot
market
decreases
from
to
).
However,
total
revenue
from
electricity
production,
including
contract
income,
is
increasing.
Figure
10
–
Total
revenue,
including
and
excluding
contract
income,
as
a
function
of
.
Figure
9
–
Total
profit
and
total
cost
as
a
function
of
the
volume
of
contracts
held
by
firm
1

29. 28
Finally,
we
define
a
spot
market
price
index
by
the
following
expression:
We
are
interested
in
the
impact
of
contracts
on
the
average
price
of
electricity
in
the
spot
market.
Spot
market
revenue
is
given
by
total
revenue
minus
contract
income,
and
market
size
is
,
the
average
demand
for
electricity
in
the
spot
market.
Figure
11
displays
as
a
function
of
.
Figure
11
–
Spot
market
price
index
as
a
function
of
.
The
evolution
of
the
price
index
is
unusual:
it
begins
at
2.71
and
decreases
until
it
reaches
a
local
minimum
of
2.70
when
GW.
Then
it
increases,
reaching
a
maximum
of
2.72
when
GW
(at
this
point,
).
Then
it
decreases
again,
reaching
a
global
minimum
of
2.865
when
Finally
the
price
index
increases
a
little,
reaching
2.695
when
GW.
If
we
look
at
the
expression
of
the
price
index,
its
behaviour
can
be
explained
partly
by
the
fact
that
total
spot
market
revenue
(the
numerator)
as
well
as
the
size
of
the
spot
market
(the
denominator)
are
decreasing.
The
rest
is
explained
by
the
change
in
slope
of
total
revenue
when
exceeds
23.7
GW.
However,
it
should
be
noted
that
these
changes
in
spot
market
price
are
small:
the
price
index
always
remains
within
1%
of
its
original
value.
In
conclusion,
the
contracts
have
little
effect
on
the
average
price
of
electricity
on
the
spot
market.

30. 29
Conclusion
In
our
model,
the
French
electricity
market
is
made
up
of
two
sectors:
a
perfectly
competitive
conventional
sector
and
a
nuclear
sector.
Electricity
demand
is
uniformly
distributed.
We
focus
our
analysis
on
market
entry
in
the
nuclear
sector.
We
begin
by
determining
the
nuclear
capacity
that
a
monopoly
would
choose
in
order
to
maximize
its
profit.
This
capacity
also
minimizes
the
total
cost
of
producing
electricity
(from
both
conventional
and
nuclear
sources)
to
meet
consumer
demand.
We
then
consider
what
happens
when
there
are
two
nuclear
producers:
a
large
firm,
the
incumbent,
and
a
small
firm,
the
entrant.
The
two
firms
compete
via
a
discriminatory
auction
mechanism
described
in
Fabra
et
al.
(2006).
When
demand
is
less
than
the
small
firm
capacity,
both
firms
sell
capacity
at
marginal
cost
and
make
zero
profit.
When
demand
exceeds
total
nuclear
capacity,
each
firm
supplies
its
whole
capacity
at
the
marginal
cost
of
conventional
producers.
When
demand
is
between
these
two
regions,
there
is
a
mixed
strategy
equilibrium.
We
find
expressions
for
each
firm’s
yearly
profit
by
integrating
over
the
distribution
of
demand.
We
then
calibrate
the
model
to
the
French
market,
assuming
that
the
nuclear
capacity
installed
on
the
market
(63
GW)
is
the
monopoly
profit-­‐maximizing
capacity.
In
the
absence
of
contracts,
the
small
firm
maximizes
its
profit
by
installing
a
capacity
of
17.5
GW.
Since
the
monopoly
capacity
is
efficient,
market
entry
leads
to
excess
capacity:
the
total
cost
of
producing
electricity
increases.
Total
profit
and
revenue
decrease,
and
the
average
price
of
electricity
drops
by
10%.
We
then
allow
the
incumbent
to
sign
long-­‐term
contracts
with
industrial
consumers
before
the
small
firm
enters
the
market.
According
to
these
contracts,
the
incumbent
supplies
a
constant
capacity
at
a
price
.
We
assume
that
–
the
contract
price
is
equal
to
the
price
of
electricity
at
the
time
the
contracts
are
signed
(when
the
incumbent
has
a
monopoly).
As
the
volume
of
contracts
increases,
market
entry
by
the
small
firm
is
reduced.
Its
profit
decreases,
while
the
incumbent’s
profit
increases.
However,
the
incumbent
cannot
recover
monopoly
profit
entirely.
Furthermore,
contracts
reduce
market
entry,
but
they
cannot
exclude
rivals
entirely
unless
the
entrant
has
large
fixed
costs.
From
a
welfare
point
of
view,
the
effect
of
long-­‐term
contracts
is
ambiguous.
On
the
one
hand,
market
entry
leads
to
excess
capacity,
so
by
limiting
entry
the
contracts
help
to
minimize
the
cost
of
electricity
production.
However,
market
entry
reduces
the
price
of
electricity,
which
may
be
viewed
as
beneficial
for
consumers.
Interestingly,
long-­‐term
contracts
do
not
have
a
significant
effect
on
the
price
of
electricity
on
the
spot
market:
it
remains
near
the
level
it
would

31. 30
have
had
with
unrestricted
market
entry.
However,
customers
who
have
signed
long-­‐term
contracts
continue
to
pay
the
monopoly
price
for
electricity.
As
a
result,
they
have
an
incentive
to
escape
the
contract
in
order
to
purchase
electricity
on
the
spot
market
instead.
An
important
extension
of
this
work
would
be
to
consider
customers’
incentives
to
sign
contracts.
We
have
assumed
that
at
the
time
of
signing,
customers
do
not
anticipate
that
there
will
be
market
entry,
or
they
do
not
internalize
the
consequences
that
the
contracts
will
have
on
a
rival
producer’s
decision
to
enter
the
market.
If
they
were
to
anticipate
this,
how
could
the
incumbent
producer
incentivise
them
to
sign
the
contract?
A
possible
answer
would
be
to
look
at
the
contract
price
.
Perhaps
the
incumbent
could
offer
customers
a
discount
at
the
time
of
signing
(setting
),
but
in
that
case
would
the
incumbent
still
benefit
from
having
the
contracts?
Similarly,
one
could
look
at
how
the
contracts
should
be
structured
in
order
to
dissuade
customers
from
ending
them
after
they
observe
market
entry
and
the
resulting
lower
prices.
A
first
step
would
be
to
examine
the
penalty
that
firms
would
be
required
to
pay
in
the
event
of
a
premature
termination
of
the
contract.

32. 31
References
Aghion,
Philippe,
and
Patrick
Bolton.
"Contracts
as
a
Barrier
to
Entry."
American
Economic
Review
(1987):
388-­‐401.
Allaz,
Blaise,
and
Jean-­‐Luc
Vila.
"Cournot
competition,
forward
markets
and
efficiency."
Journal
of
Economic
Theory
59,
no.
1
(1993):
1-­‐16.
Bessot,
Nicolas,
Maciej
Ciszewski,
and
Augustijn
Van
Haasteren.
"The
EDF
long
term
contracts
case:
addressing
foreclosure
for
the
long
term
benefit
of
industrial
customers."
Competition
Policy
Newsletter
2
(2010):
10-­‐13.
Director,
Aaron,
and
Edward
H.
Levi.
"Law
and
the
future:
Trade
regulation."
Northwestern
University
Law
Review
51
(1956):
281.
Lien,
J.
“Forward
Contracts
and
the
Curse
of
Market
Power”,
University
of
Maryland
Working
Paper
(2000)
Mahenc,
Philippe,
and
François
Salanié.
"Softening
competition
through
forward
trading."
Journal
of
Economic
Theory
116,
no.
2
(2004):
282-­‐293.
Fabra,
Natalia,
Nils-­‐Henrik
von
der
Fehr,
and
David
Harbord.
"Designing
electricity
auctions."
RAND
Journal
of
Economics
37,
no.
1
(2006):
23-­‐46.
Fabra,
Natalia,
Nils-­‐Henrik
von
der
Fehr,
and
María-­‐Ángeles
de
Frutos.
"Market
Design
and
Investment
Incentives."
Economic
Journal
121,
no.
557
(2011):
1340-­‐1360.
Rasmusen,
Eric
B.,
J.
Mark
Ramseyer,
and
John
S.
Wiley
Jr.
"Naked
Exclusion."
American
Economic
Review
(1991):
1137-­‐1145.
Segal,
Ilya
R.,
and
Michael
D.
Whinston.
"Naked
Exclusion:
Comment."
American
Economic
Review
(2000):
296-­‐309.

33. 32
Appendix
–
selected
MATLAB
code
Function
function [k2opt,profit2] = maxprofit2(f)
%MAXPROFIT2 Returns the level of capacity that maximizes firm 2's profit,
%when firm 1 has capacity k1m and holds a volume of contract f
Dmin = 33 - f;
Dmax = 78 - f;
DeltaD = Dmax - Dmin;
% k1 is the monopoly capacity, given by k1 = Dmax - b*DeltaD/(P-c)
k1 = 63 - f;
% investment cost (numeraire price)
b = 1;
% NetPrice = P - c
NetPrice = b*DeltaD/(Dmax - k1);
% maximum capacity of firm 2 - we do not want k2 to exceed k1
k2max = Dmax;
% step size (number of data points = k2max/step + 1)
step = 0.01;
% capacity of firm 2
k2vector = 0:step:k2max;
% initializing
Profit2 = zeros(size(k2vector));
NetProfit2 = zeros(size(k2vector));
for i = 1:length(k2vector)
k2 = k2vector(i);
K = k1 + k2;
% parameters for firm 2 profit
A = 3/(2*k1);
B1 = 2*log(k1/Dmin); % B1 and B2 are minus infty if f = 63
B2 = 1 + log(k1/k2);
C = -(1-b/NetPrice)*DeltaD;
if k2 <= Dmin
% profit of firm 2, before and after fixed costs
Profit2(i) = NetPrice/DeltaD*(DeltaD*k2 - B1/2*k2^2 - A/3*k2^3);
NetProfit2(i) = Profit2(i) - b*k2; % net of fixed cost
else
% profit of firm 2, before and after fixed costs
Profit2(i) = NetPrice/DeltaD*(Dmax*k2 - B2*k2^2 - A/3*k2^3);
NetProfit2(i) = Profit2(i) - b*k2; % net of fixed cost
end
end
profit2 = max(NetProfit2);
k2opt = k2vector(NetProfit2 == profit2);

34. 33
if k2opt <= Dmin
% overwrite k2opt and profit2 with analytical solution (more precise)
k2opt = (-B1 + sqrt(B1^2-4*A*C))/(2*A);
profit2 = NetPrice/DeltaD*(DeltaD*k2opt - B1/2*k2opt^2 - A/3*k2opt^3)...
- b*k2opt;
end
end
Main
script
(calls
the
previous
function)
% calculates the optimal capacity chosen by firm 2 as a function of the
% volume of contracts held by firm 1, where firm 1 has the monopoly
% capacity. Also calculates resulting profit of both firms.
Dmin = 33;
Dmax = 78;
DeltaD = Dmax - Dmin;
Davg = (Dmax + Dmin)/2;
% k1 is the monopoly capacity, given by k1 = Dmax - b*DeltaD/(P-c)
k1 = 63;
% Investment cost (numeraire price)
b = 1;
% NetPrice = P - c
NetPrice = b*DeltaD/(Dmax - k1);
% Discount = (pf - c)/(P - c)
Discount = 1;
ContractPrice = Discount*NetPrice;
% maximum volume of contracts (must be < 63)
fmax = 33;
% step size (number of data points = fmax/step + 1)
step = 0.2;
% volume of contracts
fvector = 0:step:fmax;
% initializing
k2vector = zeros(size(fvector));
Profit1 = zeros(size(fvector));
NetProfit1 = zeros(size(fvector));
Profit2 = zeros(size(fvector));
NetProfit2 = zeros(size(fvector));
TotalCost = zeros(size(fvector));
for i = 1:length(fvector)
% volume of contracts
f = fvector(i);
% updated quantities
Dmaxp = Dmax - f;
Dminp = Dmin - f;
k1p = k1 - f;

35. 34
% firm 2 capacity and profits
[k2,NetProfit2(i)] = maxprofit2(f);
k2vector(i) = k2;
Profit2(i) = NetProfit2(i) + b*k2;
% total capacity
K = k1 + k2;
Kp = K - f;
% profit of firm 1, before and after fixed costs
if k2 < Dminp
Profit1(i) = NetPrice/DeltaD*(k1p*Dmaxp + k2*Dminp...
- 1/2*Kp^2 -1/2*Dminp^2) + ContractPrice*f;
else
Profit1(i) = NetPrice/DeltaD*(k1p*(Dmaxp-k2) - 1/2*k1p^2)...
+ ContractPrice*f;
end
NetProfit1(i) = Profit1(i) - b*k1; % net of fixed cost
% total production cost minus c*Davg
TotalCost(i) = b*K + NetPrice/DeltaD*0.5*(Dmax-K)^2;
end
% total profit = firm 1 + firm 2 + coal (zero profit)
TotalProfit = NetProfit1 + NetProfit2; % after investment costs
% PriceAvg = Pavg - c (retail price index net of operating cost)
PriceAvg = (TotalProfit + TotalCost - ContractPrice*fvector)./(Davg - fvector);