Entry Deterrence in the Commons

Article (PDF Available)inInternational Economic Review 35(2):507-25 · February 1994with10 Reads
DOI: 10.2307/2527067 · Source: RePEc
Abstract
The authors analyze a common property resource model with a single incumbent firm that faces future potential entry of a rival. The cost of harvest from the resource is a function of the stock size. By drawing down current stock sufficiently, which lowers future stock, the incumbent can make entry unprofitable. The authors analyze the conditions under which the incumbent firm would deter entry and when entry would be allowed. Further, they analyze the effect that potential entry has on the harvest rate both before and after the date of potential entry and whether or not potential entry is welfare improving. Copyright 1994 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

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INTERNATIONAL ECONOMIC REVIEW
Vol. 35, No. 2, May 1994
ENTRY
DETERRENCE IN THE COMMONS*
BY
CHARLES F. MASON AND STEPHEN POLASKY1
We analyze
a
common property resource model with a single incumbent firm
that faces future potential entry of a rival. The cost of harvest from the
resource is a function of the stock size. By drawing down current stock
sufficiently,
which
lowers future stock, the incumbent can make entry unprof-
itable. We analyze the conditions under which the incumbent firm would deter
entry and when entry would be allowed. Further, we analyze the effect that
potential entry
has on
the harvest rate both before
and
after the date
of
potential entry and whether or not potential entry is welfare
improving.
1.
INTRODUCTION
On the
issue of
entry
into a common
property resource,
the literature to date has
focused
on
two
extreme
cases. The commons has
been
assumed
to be
either
open-access (no barriers to entry) or open to a fixed exogenously specified number
of
participants (infinite
barriers
to entry).
For
many
common
property resources,
there are
significant,
but
not insurmountable, setup costs. To produce from an oil
or
gas pool or an aquifer, a well
must be
drilled; to
harvest
fish or lobster, a boat
must
be purchased.
In
this paper, we study a model with positive, but finite, costs
of
entry
into the
commons. Inclusion
of
costly entry raises two important
issues
not
previously considered
in
a common property context. First, a firm that is an early
entrant, either because of historical accident or because of some special
skill
or
foresight, may
have
a
strategic advantage
as an incumbent. The threat
of future
entry may
alter the
behavior of
incumbents
and of
common
property
resource
exploitation
in
important ways. Second, equilibrium
market structure
in
exploiting
the commons can be related to the height of entry barriers.
An
example of strategic behavior by an incumbent faced
with
the threat of entry,
or
of
expansion by rivals,
is
that of
the
Hudson's Bay Company
in
furtrading
in
the
18th and
19th centuries.
In
the 19th century, the Company faced a threat from small
traders who were pushing into Company territory. The Hudson's Bay Company is
alleged
to have
responded
to these threats
in
the
following
manner:
In certain
parts
of
the
country,
it
is the
Company's policy
to
destroy forbearingg animals]
along
the whole
frontier;
and our
general
instructions recommend
that
every
effort be
*
Manuscript received May 1992.
1
We thank Ann Carlos, John Gates, Neal Johnson, Ron Johnson,
Michael
Katz, Todd Sandler,
Lou
Goudreau and Demet Haksever of the New England Regional
Fisheries
Management Council,
the
anonymous referees, and seminar participants at Boston College, Montana State University, University
of
Wyoming, Econometrics Society Winter Meetings, AERE Session
at the ASSA
Meetings,
and the
Western
Economic Association Meetings.
507
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508 CHARLES F.
MASON AND STEPHEN POLASKY
made to lay waste the country,
so as to offer no inducement to petty traders to
encroach
on the
Company's
limits.2
In
the 18th century, the
Hudson's Bay Company faced a threat
from French
traders. French furtraders
began to encroach on the area surrounding
the trading
post
at York
Factory
in
the 1730's. Either
in
reaction to
the presence of
the French,
or to dissuade further
expansion by the French, Hudson's Bay
Company beaver
harvests increased each year
from 1736 to 1740 after having been
generally
declining
in the
early 1730's. The estimated
population fell from 208,000
to 173,000
in the
time period 1736 to 1740. The
French, however, were not
dissuaded and built
an
outpost
in
the
area
in
1741.3
Elements of this type of
strategy exist
in
other examples as well.
Johnson and
Libecap (1980) indicate that
Navajo tribesman would graze a
sufficiently large
number of cattle on communal
meadows to
deter
other individuals from
also using
the grazing area. Similarly, one
can imagine firms
with
newly-acquired
leases for oil
extracting at a very fast rate so as to dissuade
others from obtaining
leases on
adjoining
lands
(Wiggins
and
Libecap 1985).
Some elements
of
the
commons
problem
also existed
prior to
the
Iraqi invasion of Kuwait
in
1990. The
Rumalia oil
field extends
across the
Iraqi-Kuwaiti
border. Control
of this
field
was a source of
tension prior to the invasion.4
Our model
integrates
strands from two
literatures, one addressing common
property resources and one
addressing entry deterrence. The central element
of
the
commons literature is that
any
one firm's actions
influence all firms' costs. This
externality
can be static
in
nature,
in
that each firm's
activities reduce its rivals'
productivity (Cornes
and Sandler
1983, Dasgupta
and
Heal 1979, Dorfman
1974,
Haveman
1973).
Or the
externality
can be
intertemporal,
in that
industry
actions
today
reduce all firms'
productivity
tomorrow
(Berck
and Perloff
1984,
Kemp
and
Long 1980, Levhari and
Mirman 1980,
McMillan and Sinn
1980).
Our model
incorporates both static and
dynamic
externalities
as
we allow the cost of
harvesting
the resource
to depend
upon
current harvest level and current stock
size,
which is a function of
past
harvest
levels. The
modern literature on
entry
deterrence stresses the
role of
precommitment
and the
strategic manipulation
of
conditions
governing
future interactions
(for
an excellent
survey
of this
literature,
see Gilbert
1989).
We
incorporate
the
possibility
of
entry
deterrence into our model
by considering
a common
property
resource
that
initially
is
exploited
by
a
single
incumbent who faces
potential
entry by
a rival
in
the
future.
These firms
may
have
market
power,
in
the sense that we allow
the demand
for
this
common
property
resource to
be
downward-sloping.
By choosing
a
high
level of
pre-entry harvest,
which
lowers
future stock
size,
the
incumbent
can credibly commit
to
entry
deterrence. Of course this
strategy
raises
the
incumbent's costs as
well;
if the
requisite
first
period
harvest is
sufficiently
great,
deterrence
will
prove
unattractive.
We solve the
model for
subgame
perfect
equilibria
and
analyze
the conditions
under which the
incumbent
would deter or
2
McLean (1849),
pp.
261-263.
3
Harvest statistics and population estimates are from Carlos and Lewis (1993).
4
In our model, we do not allow the Iraqi strategy of "forced unitization."
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ENTRY DETERRENCE IN THE
COMMONS 509
allow
entry.
We show that the
threat of entry increases
the
pre-entry equilibrium
harvest of the
incumbent over the equilibrium harvest
without threat
of
entry. Even
when the incumbent
chooses not to deter, future competition causes the
incumbent
to internalize less of
the future costs of depleting the stock.
Our result that
the threat
of
entry
increases the
harvest rate
contrasts
with
the
results of Gilbert
and Goldman (1978) and highlights the central
importance of
property rights.
In their
model, a monopoly exploiter
of an exhaustible
resource
faces
entry from
a
single
rival. Each of the two firms
has well
established
property
rights to separate
and
distinct reserve pools. Entry
is
forestalled as
long
as the
incumbent's reserves are above a
cutoff
level
(or
market
price
is below a cutoff
level) because profits for the entrant
in
the
post-entry
game
decrease
with
higher
levels of reserves held
by
the
incumbent. Similarly,
models
analyzing
the
behavior
of a dominant firm
facing
a
competitive fringe typically
find
that the
dominant
firm
retards its extraction until
fringe reserves are exhausted
(Gilbert 1978, Salant 1976).
In
contrast,
if
the resource is common
property,
then
entry
can
only
be
forestalled
by holding reserves below some
critical
level.
In
the
commons, strategic
incentives
for the
incumbent
or
dominant
firm
are to increase the
rate of
extraction when faced
with potential or
actual competition.
In addition to
analyzing
the
positive aspects of
the
model,
we
also analyze the
welfare
consequences of potential entry and entry deterrence.
A
well known
result
in
the
common-property
literature
is that the market
outcome
will
be
inefficient
unless the resource is
exploited by a single
firm.5
Models
of
entry deterrence, on the
other
hand,
often
find that the
possibility
of
entry
is
welfare-enhancing
since
it
encourages
the incumbent to raise
output.
In
our
model,
the
welfare
consequences
of
potential entry
are
ambiguous.
Potential
entry
is
good
in the
sense that
it
reduces
the
market
power
of
the
incumbent,
but bad
in
the sense that
it
increases the rent
dissipation
from
the
common
property
resource. When
the demand-side
distortions
associated
with
an unthreatened
monopolist
are
sufficiently important
relative to
the cost-side
externalities,
the
increased output associated
with
potential entry
can
lead
to a net increase
in
social welfare.
However,
if the
cost externalities are
of
paramount
importance,
or
if
entry
deterrence
entails
a
sufficiently large expansion
in
output,
then the threat of
potential entry
can reduce
social
welfare.
Indeed,
when
demand-side distortions are
sufficiently unimportant
relative to
dynamic
cost
effects,
social welfare can be
increased
by having
an unthreatened
monopolist
reduce
output.
Because the
monopolist
will
under
produce
in
the future
relative to
the social
optimum,
it
does not
take
proper account,
from a social
perspective,
of
the
effect that
increased
harvests have on future
harvest
costs.6
In
choosing
between
deterrence
and
acquiescence,
the incumbent
usually
selects the
socially
preferred strategy.
However,
the incumbent
may
find
it
profitable
to
acquiesce
5Examples of market failure with
common property resources include oil exploration and oil drilling
in a common-pool (e.g., Siegel 1985,
Wiggins and Libecap 1985), grazing
on communal
land (e.g., Johnson
and
Libecap 1980), fishing
and
hunting (e.g., Henderson and Tugwell 1979,
Paterson
and Wilen 1977).
6
The possibility that over
production due to the common property externality is exactly offset by
under production due to market power in
the output market has been analyzed
in a static
framework by
Cornes et al. (1986) and Mason et al.
(1988).
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510 CHARLES F. MASON AND STEPHEN POLASKY
when it is socially preferable to deter, even though deterrence may involve
harvesting more than is sold in the initial period.
In Section 2, we analyze a general two period model with static and dynamic
externalities and market power. We solve for the subgame perfect equilibrium and
discuss various outcomes. In Section 3, we study
the
welfare implications of
potential entry
and of
deterring
versus
allowing entry.
Section
4
contains conclud-
ing remarks.
2.
THE
TWO PERIOD MODEL
We
analyze
a two
period game
in
which an incumbent
firm is
the sole harvester
of a common
property
resource
in the first
period
but
faces
potential entry
from
a
rival firm
in the
second period.
In
the first period, the incumbent chooses a harvest
level and thereby chooses a level of stock available in
the
second period. After the
initial harvest decision by the incumbent, the entrant decides whether or not to
enter.
The entrant
must sink costs of $8F
in
period one
dollars
to enter the market
in
period two, where
8
is the discount factor between periods.
The
discount factor
measures
the
relative weight placed upon second period
effects. In
a more general
framework,
the
weighting could
be
thought of as capturing
not
only discounting
but
the relative length of the periods before and after potential entry. The relative
importance of second period outcomes can be determined
in
part by institutional
factors,
such as
licensing delays,
or
by exogenous factors
that influence the amount
of time
it
takes the
entrant to
mobilize its resources. The sunk
entry
cost can reflect
the costs associated
with
mobilizing these resources. These can represent the costs
associated
with
drilling
a well or
relocating
a
fishing
fleet
into
the
incumbent's
fishing ground,
or
institutional
factors such as the cost of
obtaining
a license or
permit
to
exploit
the
commons.
It
may
also reflect
the
opportunity
cost
of
influencing political
decisions
to
allow
entry.
Once
the
entrant is
in
the
market
it
is identical to the
incumbent
and the two
firms
compete
in
the
second
period by simultaneously choosing
levels
of harvest
and
sales. We solve the model for
subgame perfect equilibria. Throughout
the
paper
we
will
use
i
subscripts
to
denote
the incumbent's
strategy
and e
subscripts
to denote
the
potential
entrant's
strategy.
Let
St
represent
the
stock size
of the
resource
in
period t,
t
=
1,
2. Initial stock
size at
the
beginning
of
period
one
is
given
as S
1.
Let
Ht
equal
the
total harvest
in
period
t
and let
hit
and
het equal
the
period
t harvest
of
the
incumbent and entrant
respectively,
Ht
=
hit
+
het.
The
amount harvested
in
period
t
can never exceed
the
stock
size
at the
beginning
of the
period,
Ht
?
St.
Because of
rising
cost
or
falling price (or both),
the
optimizing
choice of harvest
may
be less than stock
size
even
in the
final
period.
Unless
otherwise
stated,
we
will
assume that the
optimal
harvest
choice is
less than the stock so that the stock constraint
is
not binding.
The
harvest
is
assumed to be
perishable
so that it
cannot
be stored
from period
to
period.
The
resource stock
is
linked
intertemporally by
the
growth function g(
):
St+,
=
g(St
-
Ht).
In the case of a nonrenewable
resource,
g(x)
=
x,
and in the
case of
a
renewable
resource, g(x)
>
x, g'( )
>
0,
for all x.
The cost of
harvesting
the resource for a
firm in
period
t can
depend upon
its own
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ENTRY DETERRENCE
IN THE COMMONS
511
level of harvest, the total level of harvest
of both firms in the period and stock
size.
We
assume costs can be written as
C(hjt,
Ht, St)
=
c(Ht,
St)hjt,
with
CH(H)
0,
and
CS(-)
?
0, for
j
=
i, e.
There is a static cost externality
if
and
only if
CH(H)
>
0.
There is a dynamic cost
externality from depletion
if
and only
if
CS(-)
<
0.
Let
Qt
equal total sales by all
firms in period t and let
qit
and
qet
equal
the
market sales of the incumbent and the
entrant respectively in period t,
Qt
=
qit
+
qet.
Sales are limited
by
harvests.
It is
easy
to
show that the
profit maximizing
strategy
for
every
firm in the
final period (period two)
is
to
set
harvest
and
sales
equal
and
so
we set
hj2
=
qj2,
forj
=
i, e. We do not require
the
incumbent
to sell
all
of his
period
one
harvest,
so
Q
1
H1.
The
per period
market inverse
demand
function for the sales is given by
P(Qt),
with P'( )
?
0. In a competitive
market,
where
the
sales of the firms in this
commons make up a negligible share
of total
resource sales, P'(*)
=
0,
otherwise
P'(.)
<
0. When P'( )
<
0,
we
assume
that
marginal revenue for
a firm
falls as
a rival increases sales, P'( )
+
P"(
)q
<
0,
which
implies that revenue is concave
in the firm's sales. When
P'( )
=
0,
we
assume
CH
>
0
to insure that
the
profit
function
is
concave.
These
assumptions
insure that
firms'
harvests are strategic
substitutes, i.e.,
that
reaction functions
are
negatively sloped (Bulow
et
al. 1985).
We
can
write the incumbent's
present
value of
profit
as
(1)
Hi
=
P(Ql)qil
-
c(HI,
S1)hij
+
8[P(Q2)
-
c(Q2,
S2)]qi2.
If
entry occurs, the two firms
play a Cournot game
in
which each
firm
simultaneously chooses
a
level of
harvest and sales.
Let
q
e2(S2)
and
qfY(S2)
be
the
Cournot
equilibrium
level of harvest and sales for the entrant
and
the incumbent
respectively
in
the second stage
game given stock size S2 and
let
Qd2(S2)
be
equilibrium industry output.
The entrant's
present
value of
profit
is
(2)
L1e
=
S2))- c(Q2(S2),
S2)]qe'2(S2)
-
F}
if the firm
enters;
0
otherwise.
The
entrant
will
decide not to enter
if
and only
if
his equilibrium profit
does not
exceed
the
sunk cost
of
entry,
(3)
[P(Q2(S2))
_
c(Q2(S2),
S2)]q d2(S2)
?
F.
The
way
in which
the
incumbent
can
deter
entry
is
by increasing
first
period
harvest
in
order to lower second period stock.
If
cs
<
0,
lower
stock
values
increase marginal cost and decrease
equilibrium profit
in
the
second
period
duopoly
game.7
There will exist
a critical value
of stock
size, S2,
that makes
profit
for
the
entrant
upon entry just equal to
zero.
Let
hi,
(S2)
be
the level of
first
period
harvest
that
results
in
second
period
stock size
equal
to
S2.
The mechanics
of
the
deterrence
strategy
are illustrated in
Figure
1.
Second
period
market demand
is
labeled as D.
If
the
incumbent
acquiesces,
the
resultant period
two stock yields
7
It is also possible that lower stock levels will constrain the amount of second period production since
harvest cannot be more than stock. We focus attention on the case where
cS
<
0
and do not consider
stock constraints.
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512 CHARLES F. MASON AND STEPHEN POLASKY
AC2
MC2
____ " OX____ ._____AC1
--------------------'- \
----------
MCJ
D1
D2
D
period
2
output
FIGURE
1
DETERRENCE BY INCREASED HARVEST:
THE
EFFECT ON AVERAGE
COST
AND
RESIDUAL
DEMAND
incremental costs
equal
to MC1.
The
entrant's
ex
ante
average cost,
which is
comprised
of variable cost and the
entry
fee
averaged
over its
output,
is labeled as
AC1.
The
entrant's residual demand,
which is market
demand
net
of
the
incum-
bent's
period
two Nash
equilibrium output,
is
given by
D
1 .
Now imagine
that
the
incumbent chooses to
deter, by raising
its
period
one
harvest. This lowers
period
two
stock, raising
incremental costs to
MC2
and
ex
ante average
cost to
AC2.
Finally,
with
higher marginal costs,
the incumbent's Nash
equilibrium output
following entry
is
reduced,
which
pushes
out
the
entrant's residual demand curve
to
D2. Typically,
the incumbent's
optimal
deterrence
strategy
will
result
in
a
tangency between
the
entrant's average cost and residual demand curves.
The
incumbent
will
compare
the
profits
made
by harvesting at
least
hi,
(S2)
in
period
one and
deterring entry
versus
harvesting
less
in
period
one and
allowing entry
to
occur.
We first
analyze
the case of
entry
deterrence.
Let
q[ (S2)
be the
equilibrium
level
of harvest for
the
incumbent
when it is a sole
harvester
in
period
two and has
stock size
S2.
The maximization
problem
for
the
incumbent that deters
entry by
harvesting
hi,,
which
yields
a stock
in
period
two
of
S2 s S2,
is
(4)
maxqji,hi
P(qil)qil
c(hij,
Sj)hjj
+
8[P(qij(S2))
-
c(qj(Y2),
&2)]
q'
(S2)
s.t.
qj,
?
hi,,
S2 s
S2
S2
=
g(S1
-
hi,)
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ENTRY DETERRENCE IN THE COMMONS
513
Except in instances of blockaded entry, where the profit
maximizing harvest of a
sole exploiter is greater than
hi1
(S2), the first period harvest
level will be set equal
to
hi1(S2).
The first
thing to note is that for high enough values of the
entry cost, deterring
entry
will
be
the profitable equilibrium strategy for the
incumbent.
PROPOSITION 1. Given that
cs
<
0,
there exist values
of F for which the
subgame perfect equilibrium outcome involves entry
deterrence.
PROOF. Let h/a equal the equilibrium first period harvest
of the incumbent if it
expects
to
play a duopoly game with the entrant
in
period two.
In
other
words,
hIa
is the
first
period "acquiescence" level of harvest. Period two
stock
size will be
S2
=
g(S
1
-
h/a).
Given
S2,
the
Cournot
duopoly
harvest
can be
found,
which defines
the left-hand side of equation (3). There exists a value of F
such that equation (3)
is satisfied
with
equality, i.e., the entrant is just indifferent between
entering
and
staying
out when the incumbent
harvests
hfa.
At this critical value of
F,
if the
incumbent increases first
period
harvest
infinitesimally
above
hil/,
thereby
decreas-
ing
S2
and
increasing cost,
it will
deter entry
and be a
monopolist
in
period
two
rather than a
duopolist. Doing
so would
yield
a discontinuous
jump
in
profit
so
that
deterring entry
is more
profitable
than
allowing entry.
Q.E.D.
Clearly,
the deterrence first
period
harvest
is
decreasing
in F
and
so deterrence
profits
for the incumbent are
monotonically increasing
in F
(up
until the
point
where
entry
is deterred
by following
the
monopoly
harvest
strategy).
On
the
other
hand,
acquiescence profits
do not
change
with
F. It
follows
that for
levels
of
F
above
some threshold
value,
deterrence
is
more
profitable
than
acquiescence.
In
some
cases,
the
level
of
first
period
harvest
required
to
deter
entry may
be
so
high
that first
period marginal
revenue
from selling
the
entire
harvest
is
negative.
If
there
is free
disposal,
the
incumbent
would
then
prefer
to
destroy
some of
the
harvest (excess harvest) and not bring
it to
market.8
PROPOSITION 2. An incumbent
wishing
to
deter
entry
will
destroy
some
of
the
harvest
if
and
only if demand
is
inelastic
at
hi1(S2).
PROOF. Let
A1
represent
the
Lagrange multiplier
for the constraint that
qil
?
hiI
in the
incumbent's maximization
problem
as stated
in
equation
(4). Then,
Ak
0,
(hi,
-
qil)
?
0
and
A1(hil
-
qi1)
=
0.
The constrained
first order
condition
with
respect
to
qil
is
(5)
P(
*
+
P
(
*
)qil-
k
=
?-
If
demand
is
inelastic at
hi1
(S2),
then
setting qj1
=
hi1
(S2)
will
yield negative
marginal revenue,
P(.)
+
P'(.)qi1
<
0. Since
Al
?0, equation
(5)
cannot
be
satisfied. To
satisfy equation (5),
qi1
<
hi1(S2)
must be true.
Conversely, suppose
that
qi1
<
hi1(S2).
Then
Al
=
0
and
qi1
is the
level of
8
If storage were possible, the incumbent would not destroy harvest but would store excess harvest
for future sale. Storage possibilities increase the profitability of entry deterrence in the case of excess
harvest. The inclusion of storage possibilities involves a straight-forward extension of the current model.
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514
CHARLES
F.
MASON AND
STEPHEN
POLASKY
production that
maximizes revenue. Since the revenue
function is concave
in
own
output,
it
follows
that demand
is
inelastic at
hi,
(S2).
Q.E.D.
The incumbent
engages in excess harvest much as an incumbent
monopolist can
benefit
from
holding excess capacity (Spence
1977).
A
key distinction, though,
is
that deterrence
through excess harvest can be credible
in
our
model,
since an
entrant
would
fail
to earn
positive economic profit
in
the
post-entry equilibrium,
even
when,
as
assumed
in this
model,
firms'
harvests
are
strategic
substitutes.
By
contrast,
in
a
traditional industrial organization
model,
it is not credible for
an
incumbent to deter
entry by holding excess
capacity
when firms'
outputs are
strategic
substitutes
(Bulow
et
al. 1985).
We
now
consider
acquiescence.
If first
period
harvest
is less
than
hi,
(S2),
there
will
be entry
resulting
in
a Cournot duopoly game
in
the second period.
It
is easy
to show that the
incumbent
will
set first period
sales equal to first period harvest
when it knows that
entry
will
occur so
we
let
hi,
=
qiI.
Differentiating the profit
function
for
the
incumbent, given
in
equation (1), yields
the first order
condition for
the optimal first
period harvest:
(6)
[P(qil)
-
c(qij,
SI)
+
(P'(qil)
-
CH(qij, S1))qij]
a3qe2
d
(
(p(d
+ 8g'
q2(S2)[CH
(Q2
(S2),
S2)
2-P(Q2
(S2))]
as2
+
2g
cs(Q2f(S2),
S2)qi2(S2)
=
0-
The first term
represents marginal profit
from
harvest
and sales
in
period
one.
The
second term
represents
the
change
in
second
period
profit
due to a
change
in
harvest and
sales
by
the
entrant caused by
a
change
in
stock level.
There
are two
effects:
i)
a demand side effect from
price changes
caused
by
different
levels
of sales
by
the entrant
and
ii)
a cost effect
through
the
changes
in the
entrant's
harvest
(static externality).
The
third term
represents
the
change
in
harvest cost
in
period
two caused by a
change
in
stock.
For
comparison
purposes,
consider
the case where the incumbent faces no threat
of
entry
in
period
two. Let
q'2(S2)
represent
the
incumbent's
optimal
second
period
harvest
and
sales when
it
is the sole harvester
in the
second
period given
stock
size
S2.
The
unthreatened incumbent's
optimal
first
period
harvest
and sales
is
characterized
by
(7)
[P(qil)
-
c(qij,
SI)
+
(P'(qil)
-
cH(qil,
Sl))qil]
+
8g
cs(qi(S2),
S2)qi(S2)
=
0.
As in
equation
(6),
the
first
term
represents the
marginal profit
from
first
period
harvest and
sales
and the
last term
represents
the
change
in second period
profitability
from an
increase
in
harvest
cost
caused
by
lower stock.
Comparing equations
(6)
and
(7) yields
the
next
result.
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ENTRY DETERRENCE IN THE COMMONS 515
PROPOSITION 3. Given that
Cs
<
0,
an incumbent facing entry will expand first
period equilibrium harvest as compared to an incumbent that faces no threat of
entry.
PROOF. The difference between equations (6) and (7) comes from the presence
of an additional strategic term
in
equation (6) and from having qfd instead
of
q2j
in
the last term. If c5
)
<
0,
then the last term in both equation (6) and equation (7)
will be negative. Also, because firms' harvests are strategic substitutes,
qi2(S2)
?
qij(S2).
Therefore, forequal S2,
the third term will be
greater (less negative)
in
equation (6) than
in
equation (7). Further, with c5( )
<0,
aqe2(S2)1aS2
>
0. The
sign of
the
additional strategic term
in
equation (6), 8g' (dqe2IdS2)
qi2[CH(
)
-
P'(*)], is strictly positive since CH( *-
0
or -P'(*) 2 0, with one of the two being
a
strict inequality. Because profit is a concave function
in
qj1,
it
follows that
the
value
of first period harvest which satisfies equation (7) must be smaller than the
value of
first period
harvest
which satisfies equation (6).
Q.E.D.
While not
unexpected,
this
result underscores the
importance
of both external-
ities
and strategic considerations
in
our model. The incumbent facing entry will
expand
first
period
harvest and
sales
for two reasons.
First,
the
increase
in
harvest
costs in the second
period
will
be
partially
shifted
to
a rival
firm
and will
not be
borne
entirely by
the incumbent.
Secondly, a decrease
in
stock
will
cause
the
entrant to harvest
and
sell less in the
second period.
This result also
highlights
the
central
importance
of
property rights
in
influencing
the
behavior
of an incumbent
single exploiter.
When the incumbent and
the
potential
entrant hold
separate
reserves,
under
private property, Gilbert
and Goldman
(1978)
show that the threat
of
entry
tends
to retard
the
incumbent's extraction
rate.
Proposition
3
shows
that
the reverse
effect occurs
with
common
property.
It will also be
useful to characterize
the
first
period
harvest and sales
level when
there is actual
(duopoly) competition
in
period one
as well as
in
period
two. Let
qjf
be
firm
j's equilibrium period
one
harvest, j
=
i, e,
and let
Q
d
=
qfI
+
q
e
,
qfI
>
0,
q
d
> 0.
With actual
competition
in
period one,
the first order
condition
characterizing
firm
i's
optimal
first
period
harvest is
(8)
[P(Qd)
-
c(Qd,
SI)
+
(P'(Qd)
-
cH(Q
,
SI))qj1]
+
8g
Iqe2
q
d(S2)[CH(
Qd(S2),
S2)
_
(Qd(S2))]
aS2
+ 8gcs
(Q(2
S2),
S2)qi2(S2)
=
O.
A
similar
expression
exists for
firm e.
Equation (8)
is
very
similar to
equation (6).
Indeed,
if
one
substitutes
Q
d
=
q
a
into the left-hand side of
equation (8),
where
q
a
solves
equation (6),
one obtains
(P'( )
-
CH( ))(q
d
-
q a).
This
expression
is
positive
since
P'( )
<
O
or
CH(')
>
0
and
q
d
<
Q
d -
q
a.
These observations
lead
to the
following
result.
PROPOSITION 4.
Equilibrium
harvest is
greater
when two
firms exploit
the
commons
in each
period,
as
compared
to the case where there is one
harvester in
the
first period
who allows
entry.
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516 CHARLES F.
MASON AND
STEPHEN POLASKY
The level of
first period harvest necessary to deter
entry depends upon
sunk
cost
and
can be either
greater or
less
than the quantities defined
by equations (6)
through
(8). We
will
generally
be
interested only
in
cases where
hi,
(S2)
is
greater
than
the
desired first
period
harvest when
entry is allowed
(characterized
in
equation
(6)).
If
this is not
true,
then
equilibrium
in
the model
always
involves
deterring
entry.
Therefore, we
can
rank first
period harvest
in
the
following
order. First
period
harvest with a
single incumbent facing no
threat of entry
is less than with an
incumbent that will allow
entry
in
period
two,
which
is less than with either an
incumbent that
will
deter
entry or with two harvesters
in
both
periods.
The
only
ranking that is
ambiguous is between the case
with an
incumbent that will deter
entry and the case with two harvesters in
both
periods.
In
the second
period,
harvest size is
an
increasing
function of the number
of
harvesting firms
and of
the
level of second
period stock. Deterrence
will
yield
smaller
second
period
harvests than either
allowing entry
or
having
a
single
incumbent
facing
no threat of
entry.
3.
WELFARE
The
welfare
consequences
of
potential entry
in
this
model are
complex.
There is
a cost
externality (both
static and
dynamic)
generated by
the
common
property
resource in
addition
to a
pricing distortion. Increases
in
production
in
response
to
the
threat
of
entry may
lower
price
and
increase current
consumer welfare but
may
also cause rent
dissipation
from a static
cost
externality
and
may
increase future
harvest costs
through stock depletion. The welfare
effect
of
potential entry
is
only
clear
if
there
is
just
a
cost
externality
or
if
there
is
just
a
price
distortion. With
only
a cost
externality, potential entry
causes much of the
rents that
could
have
been
attained
from
harvest
of the
resource over time to
be
dissipated
and so lowers
welfare.
With
only
a demand side
distortion,
however, potential entry
forces
the
incumbent to
expand output and so increases
welfare.
Comparing
welfare
under
deterrence and
acquiescence,
we find
that welfare is
unambiguously lower
under
deterrence than under
acquiescence
when there is
only
a
dynamic cost
externality (with no static
externality
and
no
price
externality).
Here,
the
extra
period
one
harvest associated with deterrence
only
serves
to raise
period two
costs,
which
lowers welfare.
Comparisons
of welfare
effects under
deterrence and
acquiescence
are not clear even when
there is
only
a static cost
externality or
a
price
externality.
Relative
to
acquiescence,
deterrence
yields
larger
harvests in
the first
period
but lower
stock
size and less
competition
in
the
second
period.
The
first effect
is
beneficial
(harmful)
if
there
is
only
a
price (static
cost)
externality;
the
second
effect has the reverse
sign
of the
first.
Thus,
the
net
effect
is
ambiguous.
The
discounted
flow
of social welfare with harvests
and
sales
in
period
one
(H1,
Q 1)
and in
period
two
(H2,
Q2)
can be
written as
(9)
f=
J
P(x)
dx
-
c(HI,
SI)HI
+
{
P(x)
dx
-
c(H2,
g(SI
-H))H2
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ENTRY
DETERRENCE IN THE
COMMONS 517
To facilitate further
analysis, we assume that harvest
costs are additively separable:
C(Qt, ST,
qjt)
=
IcI(Q.)
+
C2(St)]qjt.
Let
O(Qt)
=
P(Qt)
-
cl(Qt)
-
C'1
(Qt)Qt.
Then
the effect of a marginal increase in
period one harvest can be
expressed
as
(10)
=
P(Q)
-
C2(S1)
-
g'()C'2(S2)Q*2
dQi
X
{(Q*)
-
C2(S2)
aQV
C2(S2) -I
C2
(S2
)
aC2 Q *2
Further, let A
reflect the shadow cost of period one
output and let
(s
measure the
responsiveness of
period two supply to changes in
C2(S2):
aC2(S2)
(11)
A =
-
_Q_
2
=
g'(SI
-Q1)C2(S2)Q*2
dQi
1Q2V
C2(S2)
(12)
=s aC2 (S2) Q*2
Combining equations
(10), (11),
and
(12) yields
AQ
+ (Q*2 ) -
~~C2
(S2
)
(13)
0Q
=
P(Q1)
-
C2(S)-A
c2(S2)
s
-I
where
Q2
is the
equilibrium choice of harvest in period two
given period two
conditions.
Using
the above
derivations
we
can draw
several
conclusions,
which
are
given
in
the
following propositions. We begin with
a useful
benchmark result.
PROPOSITION
5. A
single incumbent without threat
of entry
will set the
socially
optimal
level
of
harvest
in both periods if and only if
P'( )
=
0.
PROOF. The welfare
maximizing level of second period harvest
will
satisfy:9
(14)
-
C2(S2)
=
0.
Using
this
fact and
equation (13),
the
welfare
maximizing
level of first
period
harvest
is Qi that solves
(15)
aQi
=
COD
-
C2(S1)
+
8A
=
O.
A
sole harvester
will
set second
period
harvest such that
(16)
P(Q2)
-
P'(Q2)Q2
-
C2(S2)
=
0.
9
Equation (14) assumes that the stock constraint on harvest does not bind. By contrast, suppose the
resource is nonrenewable, extraction costs are independent of stock size and that an optimal plan would
involve harvesting all stock by the end of the second period. Stiglitz (1976, pp. 657-659) demonstrates that
a single firm with market power will extract at the socially optimal rate if and only if
marginal
extraction
costs are constant over time
(cs
=
0) and demand, net of costs, is isoelastic.
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518
CHARLES F. MASON AND STEPHEN POLASKY
First period harvest will be
set to satisfy equation (7), which can be rewritten as
(17)
P(Ql)
+
P'(Q1)Ql
-
c2(SI)
+
8A
=
0.
Suppose that P'( )
=
0.
Comparing equations (14)
and
(16) shows that the sole
harvesting firm's second
period harvest, conditional on stock size, will be socially
optimal.
Given that second
period
harvest is
optimal, comparing equations (15)
and
(17) shows that if P'( )
=
0, first period harvest will also be socially optimal.
Conversely, suppose
that
P'( )
=$
0.
Then, comparing equations (14)
and
(16)
shows
that the
sole
harvesting firm's second period harvest, conditional on stock
size,
will
not
be
socially
optimal. Q.E.D.
When the incumbent
firm
has
some
market
power, P'( )
<
0,
it
follows
that
O(Q)
d-
c2(St)
>
0,
for both t =
1
and 2. The first-best
socially
efficient level will
not be harvested in either
period. Hence, the following welfare analysis of the
threat of
entry
is
second
best
in
nature.
One counter-intuitive result
is
that
an
unthreatened
monopolist may
harvest
more
than
is
socially optimal
in
the first
period.
To see this use
equations (13)
and
(17) to evaluate the marginal
effect on social welfare
with
respect to Q
I,
evaluated
at the incumbent's
profit
maximizing
level of
first
period
harvest:
all
,
(2
)
-
C2 (S2
)
(18) ge =
-P'(QI)Q.
+
A~s
-c2
When this
expression
is
negative,
the
monopolist
harvests more than the
socially
optimal
amount
in
the first
period.
The first
term
is
positive
because of the first
period
market
power
distortion.
However,
the
second
term,
which
captures
the
dynamic impact of period one
harvest,
is
negative.
The
dynamic
benefit
of
lower
period
one harvest is related to
increased
period
two
stocks,
which
yield
lower
period
two
costs.
Because
the
monopolist produces
less than the
socially preferred
level
in
period
two
(where
price equals marginal cost),
the
dynamic
effect is
given
less
weight
than is
socially optimal.
If the
dynamic cost
effect
dominates,
the
unthreatened
monopolist
may harvest more
than the
optimum
in
the
first
period.
This result
can
occur when the
product
of
dynamic externality
(cs),
discount factor
(8)
and the derivative of the
growth function (g') have large
absolute value. When
this is
true,
there is a
large
effect on second
period
costs from
changes
in
the first
period
harvest and second
period surplus
is
given
a
relatively large weight.
We
now turn to the
question
of
whether the
threat
of
entry
increases or decreases
welfare.
In
cases
where
the unthreatened
monopolist
harvests
the
optimal amount,
as when
P'( )
=
0,
or when it
harvests more
than the
socially optimal
amount
in
the first
period
and
deters
entry,
it
follows that
any
threat of
entry
must be
welfare-reducing. Whenever
P'(*)
is close
to zero,
the threat
of
entry
will
cause the
incumbent
to increase the harvest and
dissipate
some of the
potential
rent
that
could be
gained by
harvesting
the resource
in
an
optimal
manner.
When
the
unthreatened
monopolist
harvests more than
is
socially optimal
in
the
first
period,
the threat
of entry causes an increase
in
first
period harvest,
which
lowers social
welfare.
If
the monopolist deters
entry,
then it remains a
monopolist
in
the second
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ENTRY DETERRENCE IN THE COMMONS
519
period so that there is no reduction in the second period distortion. We
summarize
this discussion in the following proposition.
PROPOSITION 6. There exists
a
set of parameters under
which the threat
of
competition must lower social welfare.
As noted above, one scenario where the threat of entry is welfare
reducing
occurs
when the
unthreatened monopolist
harvests more than
the
socially preferred
level. In this case, deterrence increases over exploitation and
so
lowers
welfare. It
is
interesting to contrast
this
result
with the
corresponding
result in
Gilbert and
Goldman
(1978). They also
find
that the threat
of entry
is
welfare reducing
but
for very
different reasons. Because
entry
is
forestalled
in their framework
by
holding
reserves
above a certain level, the threat
of
entry tends to reduce
the incumbent's rate of
activity.
In
most models
of
nonrenewable
resources, however,
a
monopolist
will tend
to
under exploit the resource relative to a social planner. Hence,
the threat of
entry
is
welfare reducing because it further retards the pace of extraction. As
we noted in the
introduction, the key distinction between the Gilbert and Goldman model
and
ours
is
the
nature
of
property rights:
in
their
model,
the firms
exploit
different
deposits
of the
resource;
in
ours, the firms exploit the same deposit.
In
some
sense
then,
incentives
in
the
two models are reversed. Hence, our Proposition 6 might
be viewed as a
common-property analog
of Gilbert and Goldman's welfare
results.
One
key
difference between Gilbert and Goldman's
welfare
results
and
ours
is
that
our results
are not
unambiguous.
If
the
price
distortions are
sufficiently
important, then for sufficiently large values of
F,
the threat of entry
must be
welfare-enhancing.
PROPOSITION 7.
If P'( )
<
0,
then
there exists
a
set
of parameters
under
which
the
threat
of competition
must raise social
welfare.
PROOF.
For
sufficiently large
values of
F,
entry
is
blockaded.
Take the smallest
such
value;
call
it F.
For
values
of
F
just
a bit
smaller than
F,
the deterrence
harvest is a
bit larger
than
the monopoly harvest,
and
entry
will
surely
be deterred.
Hence,
this
gives
the
same number of firms
in
period
two as
the
unthreatened
single-firm exploitation,
with
slightly larger
first
period output.
When the right-hand
side
of
equation (18)
is
positive,
this
effect will
cause
an increase in social welfare.
Q.E.D.
In
a related vein, an incumbent threatened by entry need
not select the socially
optimal
alternative between
acquiescence
and deterrence.
If
dynamic
cost effects
are
very important,
the
period
one harvest
required
to deter
entry
can
be
relatively
close to the
monopoly
level. If these
dynamic
effects are
large
relative to
demand-side distortions and second period effects
are
sufficiently
valuable relative
to
first
period effects,
then deterrence can be
privately optimal
but
socially subopti-
mal.10 This
corresponds to
the
traditional finding
in
the Industrial
Organization
10
An example of the kind of conditions that would yield this scenario may be fishing for ocean perch
or halibut. The population of halibut and ocean perch grow slowly so that
it
is difficult for
the
population
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520
CHARLES
F. MASON
AND STEPHEN POLASKY
SWA
tA
SWN
N
,
ON
N *
Si
S1
initial
stock
size
FIGURE
2
SOCIAL VS.
PRIVATE
OPTIMA:
THE ROLE
OF INITIAL
STOCK
SIZE
literature (Gilbert
1989).
If dynamic
cost
effects
are slight,
however,
deterrence
may
require
a
large period
one
harvest.
If demand-side
distortions
are important
relative to
the
dynamic
cost
effects
and
second
period
effects
are
sufficiently
valuable relative
to
first
period
effects, it is
possible
that deterrence
is privately
suboptimal
but
socially
optimal.11
This
possibility
is illustrated
in Figure
2.
In
Figure
2,
we compare
the
incumbent's
profit
under
deterrence
(HN) and
acquiescence
(HA) and
social
welfare
under
deterrence
(SWN)
and
acquiescence
(S
WA)
over
a
range
of initial stock
sizes
(S
1).
Acquiescence
profits
and welfare
are
increasing
in the
level of
initial
stock
because
increases
in
stock
size yield
lower
harvesting
cost.
For deterrence,
the incumbent
must
drive stock
down
to S2,
which
fixes
second period
profit,
regardless
of
initial
stock
size. Thus,
larger
values
of
SI
require
larger amounts
of
period
one harvest
to
deter
entry,
without
yielding
second
period
gains.
At