❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s
❊s❝♦❧❛ ❞❡
Pós✲●r❛❞✉❛çã♦
❡♠ ❊❝♦♥♦♠✐❛
❞❛ ❋✉♥❞❛çã♦
●❡t✉❧✐♦ ❱❛r❣❛s
◆◦ ✹✶✶ ■❙❙◆ ✵✶✵✹✲✽✾✶✵
❈❤❛s✐♥❣ P❛t❡♥ts
❋❧❛✈✐♦ ▼❛rq✉❡s ▼❡♥❡③❡s✱ ❘♦❤❛♥ P✐t❝❤❢♦r❞
❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s
♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛
❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳
❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦
❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐
▼❛rq✉❡s ▼❡♥❡③❡s✱ ❋❧❛✈✐♦
❈❤❛s✐♥❣ P❛t❡♥ts✴ ❋❧❛✈✐♦ ▼❛rq✉❡s ▼❡♥❡③❡s✱ ❘♦❤❛♥ P✐t❝❤❢♦r❞ ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵
✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✹✶✶✮
■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳
Chasing Patents
¤
Flavio M. Menezes
EPGE/ FGV
and
School of Economics
Australian National University
Canberra, ACT , 0200
Aust ralia
Rohan Pitchford
RSSS and NCDS
Australian National University
Canberra, ACT , 0200
Aust ralia
March 1, 2001
A bst r act
We examine t he problem faced by a company t hat wi shes to pur-chase pat ents in t he hands of two di¤erent patent owners. Comple-ment arity of t hese pat ents in the product ion process of t he company is a prime e¢ ciency reason for t hem being owned (or licenced) by t he company. We show that t his very same complement arity can lead t o patent owners behaving strat egically in bargaining, and delaying t heir sale t o the company. When t he company is highly leveraged, such ine¢ cient delay is limit ed. Comparat ive statics result s are also obtai ned. Relevant applicat ions include assembly of pat ent s for drug t reat ment s from the human genome, and land assembly.
K ey -wor d s: pat ents, complementarit y, bargaining. J EL C lassi…cat ion: C78, O31, O34.
1
I nt r oduct ion
The lit erat ure on asset ownership (Hart and Moore (1990) and Hart ( 1995)),
predict s t hat complement ary asset s should be owned by t he same part y.
While seemingly simple, t he reason behind t his result is quit e subt le: T wo
asset s are complement ary if t he marginal product of asset -speci…c invest
-ment s by any one part y is zero if t he asset s are not used t oget her. Separat e
ownership of t hese asset s weakens each party’s out side value of t he asset , and
t his increases t he surplus t hat is ext ract ed by t he opposing part y. Ant
ici-pat ing such a problem, each party will invest less in t he speci…c project t han
is desirable. I n cont rast , if t he asset s are owned t oget her, t hen t he owner’s
out side value of t he combinat ion is high. Such an owner has great er incent ive
t o invest ex-ant e.
This result assumes t hat t here is a compet it ive market in asset s prior t o
part ies agr eeing t o do business wit h one-anot her. However, t here are many
cases in which market s for asset s can be quit e t hin. Two (or more) complet ely
separat e owners will oft en make separat e discoveries t hat would – purely
t hrough chance – generat e higher value if t hey were combined. Combining
such discoveries t akes on an import ance of it s own when market s are t hin: An
int erest ing example of t his was t he development of a virus-resist ant papaya
in Hawaii. To be able t o produce and dist ribut e t r ansgenic seeds resist ant t o
t he papaya ring spot virus, it was necessary t o obt ain t he legal right s of ot her
pat ent s t hat would be infringed. Four relevant pat ent s had t o be ident i…ed
and t heir owners cont act ed for negot iat ions. It t urned out t hat “ t he process
of obt aining agreement s from t he pat ent holders was long and arduous. Each
pat ent holder had his own agenda for licensing, ranging from having lit t le
t he most favorable deal.” 1
The problems encount ered wit h papaya are likely t o be minor when
com-pared t o t he recent ly-mapped human genome. M any DNA sequences have
already been pat ent ed. I n fact , t he U.S. Pat ent and Trademarks O¢ ce had
issued more t han 2,000 pat ent s covering gene sequences by t he end of 1999.
New applicat ions will almost cert ainly require mult ipleDNA sequenceswhose
pat ent s are held by di¤erent owners. Will t his pat ent assembly be
accom-plished e¢ cient ly or will it hinder innovat ion and t he discovery of new drugs
or t reat ment s? This quest ion is t he basis of our paper .2
We develop a model t o capt ur et he pr oblem of combining separat epat ent s
(or ot her asset s such as land) when owners can delay sale for st rat egic
ad-vant age. Our main result is t hat complement arit y – while a major reason
for asset s t o be owned t oget her – is also more likely t o lead t o cost ly delay
in pat ent purchase. Grossman and Hart (1980) analyze a similar problem;
a raider will not t akeover a corporat ion because t he ret ur ns fr om any
cor-porat e management improvement s int roduced by t he raider will be capt ured
by exist ing shareholders. The r eason for such ine¢ ciency is t he public goods
nat ure of managing a corporat ion. Grossman and Hart t hen examine several
devices t hat are meant t o avoid t his nonexcludibilit y problem. In cont rast ,
ine¢ ciency in our set t ing arises from st rat egic behavior by pat ent owners.
The land assembly pr oblem has also some common feat ures wit h t he
chasing pat ent s game we analyze; a developer want s t o assemble several
1A s cit ed in “ V irus-Resist ant Papaya in Hawaii: A Sucess St ory,” ISB News
Re-port , January 1999, available at www.plant .uoguelph.ca/ safefood/ archives/ agnet / 1999/ 1-1999/ ag-01-09-99-01.t xt .
2T he pr oblem of pat ent assembly has been recognized in many areas. Lowe (2000), for
example, suggest s t hat “ for invent ions involving mult iple pat ent s held by di¤erent par t ies, t here are high t ransact ion cost s associat ed wit h bargaining over right s, which can lead t o
blocking of commercial development ” in t he healt h care indust ry. His example is t he
parcels of land t o undert ake a project t hat delivers posit ive ext ernalit ies t o
surrounding land owners. When t he developer cannot make a credible
all-or-not hing o¤er, ine¢ ciency is likely t o occur: Exist ing owners will wait in or der
t o capt ure t he rent s result ing from t he complet ion of t he project . 3 In our
model t he source of ine¢ ciency is t he bargaining process, not t he exist ence
of posit ive ext ernalit ies from project complet ion.
While t he st andard lit erat ure on pat ent s4 focuses on t he link bet ween
R&D and pat ent s, we abst ract from t he development st age. T his capt ures
value of combining pat ent s t hat may be generat ed purely t hrough
serendip-ity – and unant icipat ed by t he separat e developers. Our goal is t o st udy t he
mechanism by which mult iple pat ent s are acquired and invest igat e it s
impli-cat ions for e¢ ciency. In our model t her e are t wo pat ent owners, and a t hir d
party who wishes t o combine t hem. Each pat ent owner can choose t o
nego-t ianego-t e sale of nego-t he panego-t ennego-t nego-t o nego-t he nego-t hird parnego-t y immedianego-t ely, or delay negonego-t ianego-t ion
in t he hope of a bet t er deal. The model is described in det ail below.
2
T he M odel
There are t hree players in t he model. A pharmaceut ical company (player 0)
want s t o buy t wo pat ent s, and realize a value v from owning t he ent ire set .
However, each of t hese pat ent s are owned by players – t wo pat ent owners –
3Grossman and Hart ar gue t hat t hese ext ernalit ies could be avoided if t he developer
could hide his int ent ions from t he lot owners. T here is however a large lit er at ure on land assembly. Recent papers eliminat e t he ext ernalit y problem eit her by assuming t hat lot owners can make …nal o¤ers above t heir reservat ion prices, as in Eckar t (1985), or by t aking a cooperat ive appr oach, as in A sami ( 1988). O’Flahert y (1994) st udies urban renewal – when a public aut hor it y has t he power t o buy t he lot s and resell t hem t o t he developer – and shows t hat it is not a good remedy for t he ext ernalit y pr oblem.
4A rrow (1962) is a classical r efer ence on analyzing invent ions t hat r educe pr oduct ion
1 and 2 respect ively. Player i = 1; 2 values it s pat ent at wi. The
pharma-ceut ical company values t he individual pat ent of i at vi; i = 1; 2. We assume
t hat t he value of t he t wo asset s t oget her exceeds t he sum of t he individual
valuat ions, i.e.
v > v1+ v2;
and t hat t he pharmaceut ical company values t he individual pat ent s at least
as much as t he owners, i.e.
vi ¸ wi:
Ideally, t he pharmaceut ical company would like t o engage each of t he
owners t oget her, make a t ake-or-leave-it o¤er wi; i = 1; 2, and realize t he
value v ¡ w1¡ w2. This may not be possible. A pat ent owner may perceive
an advant age from not going t o t he bar gaining t able when t he ot her owner
is present . In ot her words, it might be advant ageous for an owner t o delay
sale, perhaps hoping for a higher price lat er on.
To model t his possibilit y, we assume t here are t wo possible t imes at which
each par ty can go t o t hebargaining t able, ti = n (“ now” ) and ti = l (“ lat er” ),
i = 1; 2. We assume t hat t he pat ent owners i = 1; 2 simult aneously and
non-cooperat ively choose pi t he probability of going t o t he bargaining t able
now, wit h probabilit y (1 ¡ pi) of going lat er. This choice leads t o four
pos-sible event s: Bot h part ies are at t he bargaining t able now, probabilit y p1p2;
party 1 is at t he t able now, and party 2 lat er and vice-versa, probabilit ies
p1(1 ¡ p2) and p2(1 ¡ p1), and; bot h part ies are at t he bargaining t able lat er,
probability (1 ¡ p1) (1 ¡ p2).
We assume t hat once players ar e at t he t able, t hey bargain e¢ cient ly over
t he exchange of pat ent s. This allows us t o examine t he pur e quest ion of how
st rat egic avoidance of bargaining a¤ect s welfare, wit hout biasing result s by
Nash bargaining t o det ermine t he payo¤s t o each player in each event . We
assume t hat t he payo¤ t o an individual in a bargain is generically as follows:
payo¤ =
(t hreat point payo¤ ) + (bargaining share)¢[available surplus - sum(t hreat point payo¤s)].
(1)
Our int erpret at ion t hroughout of t he t hreat point payo¤ of a bargainer is
st andard: it is t he payo¤ if bargaining breaks down complet ely, wit h no
pos-sibilit y of reconciliat ion. Thus, t he overall payo¤ is t he sum of t he t hreat point
payo¤, and a share of t he gains from t rade.
We assume t hat t he pharmaceut ical company is unable t o commit t o
leave t he bargaining t able at t ime n. To do so would not be subgame perfect :
Speci…cally, suppose t hat t he company purchased a pat ent fr om one owner at
dat e n. This agreement yields a posit ive payo¤, and becomes sunk. The t ime
l agreement also yields a posit ive payo¤, and t he pharmaceut ical company
has an incent ive t o st ay at t he t able. Therefore, we only need t o focus on t he
payo¤s of t he two pat ent owners, since only t hese players are able t o make
st rat egic choices in t he model.
Let t he not at ion s (tj; tk) denot e t he payo¤ t o eit her player j 2 f 1; 2g or
k 6= j 2 f 1; 2g, when t he out come of t heir choices of pj and pk are tj 2 f n; lg
and tk 2 f n; lg respect ively. I f all t hree players ar e at t he bargaining t able
at t ime n, so t hat t1= t2= n, t hen i = 1; 2 receive
s (n; n) = wi + ®i ¢(v ¡ w1¡ w2)
in present value dollars, where ®j ¸ 0, §2
j = 0®j = 1, is t he bargaining share
of t he gains from t rade v ¡ w1¡ w2 of player j = 0; 1; 2, in a t hree player
placed on t he next best use of t he pat ent . We assume t hat dat e l payo¤s are
discount ed by t he fact or ± 2 (0; 1), so t hat t he payo¤ t o player i = 1; 2 in
present value t erms from t he t hree player bargain is
s (l; l) = ± ¢(wi + ®i ¢(v ¡ w1¡ w2)).
We can int erpret a higher ± as a longer period of delay bet ween dat es n and
l, or direct ly as a st ronger t ime preference for all players.
Det er minat ion of t he payo¤s for sit uat ions where t here is only one pat ent
holder at t he t able at any given t ime, requires some careful t hought . Suppose
pat ent holder j is at t he t able at dat e l. At t his t ime, pat ent holder k 6= j
has made a bargain wit h t he pharmaceut ical company. T herefore t he t ot al
available surplus at dat e l is v. However, t he company can t hreat en not t o
purchase j ’s pat ent , and just use t he …rst owner’s pat ent , i.e. t he company’s
t hreat point payo¤ is v1. Applying t he Nash bargaining formula (1) above,
yields a present value payo¤ t o player j of
s (l; n) = ± ¢(wj + ¯j ¢(v ¡ vk¡ wj)) ,
where ¯j 2 (0; 1) is t he bargaining share of player j vis-a-vis t he company.
Not e t hat t he company is pot ent ially advant aged because it can ext ract a
t hreat point payo¤ of vk > 0 in t his bargain, due t o t he fact t hat it now holds
pat ent k.
Now suppose j is at t he bargaining t able at dat e n, and k negot iat es at
dat e l. Consider t he agreement bet ween owner j and t he company at dat e
n. The company’s t hreat – should bargaining br eak down – is not t o deal
wit h player j , wait unt il dat e l, and receive payo¤ (1 ¡ ¯k) ¢(vk¡ wk), being
it s share 1 ¡ ¯k of t he gains from t r ade vk ¡ wk in t he deal wit h t he ot her
ant icipat e t hat e¢ cient bargaining will yield t he t ot al value v.5 This yields a payo¤ at dat e n t o player l of
s ( n; l) = wj + ¯j ¢[v ¡ wj ¡ ± ¢(1 ¡ ¯k) ¢(vk ¡ wk)].
The import ant point t o not e is t hat t he pharmaceut ical company’s fut ure
payo¤ from dealing wit h player k alone, a¤ect s t he surplus in t he bargain
at dat e n wit h player j . T hus, t he payo¤s capt ure –in a rigorous way–
int ert emporal compet it ion bet ween t he two pat ent -holders.
Now consider pat ent owner j0s choice of pj at t he beginning of t he game.
The owner ’s expect ed payo¤ is calculat ed by weight ing t he payo¤s derived
above wit h t he pr obabilit ies of each event :
¼j = pj ¢pk ¢[wj + ®j ¢(v ¡ wj ¡ wk) ] (2)
+ pj ¢(1 ¡ pk) ¢[wj + ¯j ¢(v ¡ wj ¡ ± ¢(1 ¡ ¯k) ¢(vk ¡ wk))]
+ (1 ¡ pj) ¢pk¢[± ¢( wj + ¯j ¢(v ¡ vk ¡ wj) )]
+ (1 ¡ pj) ¢(1 ¡ pk) ¢[± ¢(wj + ®j ¢( v ¡ wj ¡ wk))]
3
Solut ion and R esult s
We can derive Nash equilibria in t he model by examining t he derivat ive of
(2). Aft er some simpli…cat ion, t he derivat ive becomes:
d¼j
dpj
= pk¢X + ( 1 ¡ pk) ¢Y (3)
where
X = s (n; n) ¡ s (l; n) (4)
= (1 ¡ ±) ¢wj + (®j ¡ ±¯j) ¢(v ¡ wj ¡ wk) + ± ¢¯j ¢(vk¡ wk) ,
5T hat is, bot h players know t hat bargaining is e¢ cient , and t hat bargaining at dat e 2
and
Y = s (n; l) ¡ s (l; l) (5)
= (1 ¡ ±) ¢wj + ( ¯j ¡ ± ¢®j) ¢(v ¡ wj ¡ wk)
+ ¯j(wk ¡ ± ¢(2 ¡ ¯k) ¢(vk¡ wk))
Pr oposit ion 1 There is ine¢ cient delay in equilibr ium if X < 0 and Y > 0
in the for m of multiple equilibria ( p1; p2) 2 f (1; 0) ; (0; 1) ; (p1; p2) g with
pk = (1 ¡ ±) ¢wj + (¯j ¡ ±®j) ¢(v ¡ wj ¡ wk) + ¯j ¢(wk ¡ ± ¢(1 ¡ ¯k) ¢(vk¡ wk))
(1 + ±) ¢(¯j ¡ ®j) ¢(v ¡ wj ¡ wk) + ¯j ¢(wk ¡ ±(2 ¡ ¯k) ¢( vk¡ wk))
(6)
j 6= k = 1; 2.
Pr oof. > From equat ion (3) , we have d¼j
dpj = 0 where pk = pk =
Y Y ¡ X 2
(0; 1) as X < 0, Y > 0. On subst it ut ion, pk is given by equat ion (6). T he
best response corr espondences of t he owners are given by
pj =
8 <
:
0 for pk > pk
[0; 1] for pk = pk
1 for pk < pk
j 6= k = 1; 2.
This is because if pk > pk, d¼j
dpj = pk ¢X + (1 ¡ pk) ¢Y < 0, as X < 0; Y > 0.
Similarly, if pk < pk, d¼dpjj > 0 as X < 0; Y > 0. To calculat e equilibria,
suppose …rst t hat p2 = 0 < p2. Owner 1’s best response is p1 = 1. Owner
2’s best response t o p1 = 1 > p1 is p2 = 0. Thus, (1; 0) a Nash equilibrium,
as is (0; 1) by a symmet rical argument . Consider t he point (p1; p2). Neit her
owner increases t heir payo¤ from deviat ing, so t hat t his point is also a Nash
Equilibrium. There are no ot her Nash equilibria, since owner 2 will deviat e
from any point (p1; p2) if p16= p1.
First not e t hat delay is always ine¢ cient , because t he t ot al available
by not ing t hat owners 1 and 2 are playing an int ert emporal coordinat ion
game. T he t erm X is t he di¤erence between owner j ’s payo¤ from
bargain-ing now and bargainbargain-ing lat er, condit ional on owner k bargainbargain-ing now (i.e.
X = s (n; n) ¡ s (l; n)). Since X < 0, owner j pr efers t o be absent now when
owner k is present . The t erm Y is t he di¤erence bet ween j ’s payo¤ from
bargaining now and bargaining at t ime l when owner k bargains at t ime l
(i.e., Y = s (n; l) ¡ s (l; l)). Owner j pr efers t o bargain now in t his case. I n
summary, bot h owners would prefer t o be absent from t he t able if t he ot her
player is present ; t hey wish t o coordinat e t o be apart .
The proposit ion as it st ands does not give us su¢ cient insight int o t he
basic mot ivat ion for equilibrium delay; we need t o examine t he st ruct ure of
payo¤s more carefully:
Pr oposit ion 2 Suppose that discounted bilateral bargaining yields a larger
share of sur plus than trilateral bargaining (i.e. ±¯i > ®i, i = 1; 2). Then there
is ine¢ cient delay in t he equilibr ium outcome if patents are ( su¢ ciently)
complementar y, i.e. if either
( i) vi = wi, and ± is su¢ ciently near unity ( precisely, ± > wwii+ ®+ ¯ii( v¡ w(v¡ wii¡ w¡ wkk));
i 6= k = 1; 2); or
( ii) v is su¢ cient ly large.
Pr oof. For case (i), we have X negat ive for ± > wi+ ®i(v¡ wi¡ wk)
wi+ ¯i( v¡ wi¡ wk) and Y
posit ive if ±¯j ¡ ®j > 0. For (ii), ®j ¡ ±¯j < 0 implies from equat ion (4) t hat
X is monot onic decreasing in v, so t hat X < 0 for v su¢ cient ly large. Since
¯j ¡ ±®j > 0, we see from equat ion ( 5) t hat Y is monot onic incr easing in v,
so Y > 0 for su¢ cient ly large v. From Proposit ion (1), t here is delay in all
The int uit ion for Proposit ion 2 is as follows. For part (i), t he
pharma-ceut ical company’s value of a single pat ent is no great er t han t he value t o
t he owner (vi = wi). T hus, t he pharmaceut ical company does not gain much
of an advant age if it purchases a pat ent . This is t rue when t he company is
bargaining wit h only one owner at eit her dat e n or at dat e l. In t he former
case – a deal wit h one owner at dat e n – t he company ant icipat es t hat it does
not have much int ert emporal bargaining power from a fut ure deal. In t he
lat t er case – an agreement wit h one owner at dat e l – t he company holds a
pat ent t hat doesn’t give it much immediat e bargaining power. T he lack of a
st rong t hreat point on t he part of t he company when t here is only one owner
at t he t able, means t hat t he owner is negot iat ing over a larger net surplus.
I n addit ion, t he ant icipat ed share of t his net surplus t o an owner (when t he
ot her is absent ) is larger, even when t he owner must delay it s going t o t he
bargaining t able ( i.e. we assumed ±¯j > ®j). Consequent ly, bot h part ies
would prefer t o be at t he t able alone. From proposit ion 1, t her e is delay in
all t hree equilibria. Wit h part (ii), t here is ine¢ cient delay for analogous
reasons. The t ot al available surplus v is high, t herefore t he gains t o owners
from being alone is also high.
In bot h cases (i) and (ii), t he driving force behind delay is t he degree
of complement arity and t he fact t hat bilat eral bargaining power exceeds t
ri-lat eral bargaining power. Since v > v1 + v2, and ±¯j > ®j, each pat ent
owner has an increased incent ive t o not coor dinat e wit h t he ot her owner. I n
t his way, t he part ies can “ divide and conquer” : By negot iat ing separat ely
– at least wit h some probabilit y – t here is a sunk component t o t he dat e n
agreement . The owners seize a larger share ±¯j > ®j of a lar ge gain from
t rade. For example, if party 1 bargained at dat e n, and received share ¯1,
more of an expect ed pie t han if t hey negot iat e at t he same dat e wit h shares
®1 and ®2.
4
C om par at ive St at i cs
I n t his sect ion we examine t he changes in equilibrium behavior result ing from
changes in some of t he st ruct ural paramet ers. To do so we writ e t he …rst
-order condit ion as a funct ion of pk and of t he exogenous variables. T hat
is,
d¼j
dpj
= f (pk; £ ) ;
where £ = (±; v; vk; vj; wj; wk) is a vect or of exogenous paramet ers.
Not e t hat f (pk; £ ) is decr easing in pk under t he assumpt ions in
propo-sit ion 1, since fpk = X ¡ Y < 0. Thus, t he change in t he equilibrium value
of pk t hat result s from a change in t he paramet er µ 2 £ depends on how
f changes wit h respect t o µ. For example, if we can show t hat f (pk; £ )
increases wit h µ for all values of pk; t hen we can argue t hat t he equilibrium
value of ¹pk increases wit h µ as well. This is depict ed in …gur e 1 wit h some
abuse of not at ion where for convenience we writ e f (pk; µ).
In …gure 1 if a rise in paramet er µ leads t o a rise in f , t hen ¹pk rises t o
¹p0k, and if it leads t o a fall in f , t hen ¹pk falls t o ¹p00k. That is, we only need t o
Figure 1: Comparative Statics
k
kp
p, ?pk,??
f
?pk,??
f ''
k
p pk pk'
As a check on int uit ion, consider t he e¤ect of an increase in ±. We would
expect t hat t his leads t o an incr ease in delay. Di¤er ent iat ing f wit h respect
t o ± yields
f± = ¡ 2wj ¡ (®j + ¯j) (v ¡ wj ¡ wk) ¡ ¯j (1 ¡ ¯k) (vk¡ wk) < 0
as expect ed: When ± rises, t hen not only does t he gain from fut ure payo¤s
rise, but t hepharmaceut ical company’st hreat point payo¤ from not t o dealing
wit h j at t ime n is improved (i.e. s (n; l) falls).
The comparat ive st at ics of v are not as direct . Di¤erent iat ing f wit h
respect t o v yields:
fv = pk( 1 + ±) (®j ¡ ¯j) + ¯j ¡ ±®j (7)
which is of ambiguous sign under t he assumpt ion t hat discount ed bilat eral
bargaining power exceeds t rilat eral bargaining power (±¯j > ®j). Figure 2
Figure 2: Comparative Statics of v
k p )
, (p v fv k
k p
) , (p v fv k j j ?? ? ? j j ?? ? ? 1 0 rises
pk pkfalls
^
When ¹pk = 0, fv = ¯j¡ ±®j > 0, and when ¹pk = 1, t hen fv = ®j¡ ±¯j < 0.
Also, fv is decreasing in pk. I t follows t hat fv is increasing below t he value
^
pk where fv = 0, and is decreasing above ^pk. Therefore, ¹pk is increasing for
¹pk < ^pk and decreasing for ¹pk > ^pk. Using t he de…nit ion of ^pk allows us t o
derive t he following proposit ion.
Pr oposit ion 3 Suppose that discounted bilateral bargaining yields a larger
share of surplus than trilateral bargaining ( i.e. ±¯i > ®i, i = 1; 2). Then
ine¢ cient delay decreases (increases) with v if
¯j ¡ ±®j
(1 + ±) (¯j ¡ ®j)
> (< )
(1 ¡ ±) ¢wj + (¯j ¡ ±®j) ¢(v ¡ wj ¡ wk) + ¯j ¢(wk¡ ± ¢(1 ¡ ¯k) ¢(vk¡ wk))
Similar comparat ive st at ics exercises can be accomplished for t he
remain-ing exogenous par amet ers in order t o obt ain speci…c condit ions under which
increases in any of t he variables f vk; vj; wj; wkg may lead t o an increase (or
decrease) in ine¢ cient delay. We omit t he det ails.
5
D iscussion and Ex t ensions
Our aim in t his paper is t o provide a simple bargaining framework t o analyze
t he problem faced by a company who want s t o buy complement ar y pat ent s
from dist inct pat ent owners. Accordingly, several ext ensions are possible and
some are discussed below.
5.1
W ealt h C onst r aint s and E¢ ciency
I n t he analysis above, it was impossible for t he company t o commit – at dat e
n – not t o negot iat e at dat e l. Being able t o commit not t o bargain is an
ext reme version of a commit ment t o be a t ough negot iat or. T he presence of
wealt h const raint s on t he company admit s t he possibility t hat it can credibly
commit t o be a t ougher negot iat or at dat e n. Weexplore t his possibility here.
Suppose t hat t he pharmaceut ical company has at most wealt h W 2 [w1+
w2; v] t o expend on t he purchase of t hepat ent s. T his is possible, for example,
if t he company is su¢ cient ly highly leveraged. Limit ed wealt h means t hat
t he company can o¤er at most W for t he two pat ent s in any agreement s wit h
t he owners. For illust r at ive purposes, suppose t hat wealt h is t he minimum,
at W = w1+ w2. Consider t he bargaining out come in each event . When bot h
owners negot iat e at dat e n, t hey can receive no more t han w1+ w2bet ween
t hem. Since we assume t hat bargaining is e¢ cient (t o focus exclusively on
t he st rat egic incent ives t o delay), t he payo¤ for each part y is s(n; n) = wi
Now suppose t hat only owner 1 is present at dat e n. The company’s
t hreat point payo¤ is t he amount it will receive from a dat e l deal wit h par ty 2.
I n t his circumst ance, 2 receives ± ¢min f w1+ w2; w2+ ¯2(v2¡ w2) g.
There-fore, in t he dat e n agreement wit h t he company, part y 1 receives
s (n; l) = min [w1+ w2; w1+ ¯1(v ¡ w1¡ ± ¢min f w1+ w2; w2+ ¯2(v2¡ w2)g)] .
Consider t he case where part y 1 bargains at dat e l – aft er player 2 reaches
agreement at dat e n. T he wealt h remaining t o t he company for bargaining
purposes is w1+ w2¡ s (l; n) · w1 (where s (l; n) is de…ned symmet rically t o
s (n; l) – wit h subscript s 1 and 2 swit ched). It follows t hat t he payo¤ t o 1 is
0. Finally, if bot h part ies bargain at dat e n, t hey receive s (n; n) = wi. This
gives t he following expect ed pro…t t o player 1:
¼1= p1¢p2¢w1
+ p1¢(1 ¡ p2) ¢s (n; l)
+ (1 ¡ p1) ¢(1 ¡ p2) ¢w1.
Di¤er ent iat ing gives
d¼1
dp1
= p2w1+ (1 ¡ p2) (s (n; l) ¡ w1) > 0.
Therefore, t here is no ine¢ cient delay in equilibrium. The wealt h const raint
serves t o commit t he company t o be a hard bargainer.
Clearly from t he reasoning in t he …rst part of t he paper, if W = v,
t here is ine¢ cient delay. By t he cont inuit y of payo¤s, t here must be some
level of wealt h such t hat ine¢ cient delay is eliminat ed. The presence of a
wealt h const raint on t he company can act as a credible commit ment t o t ough
by st rat egic bargaining behavior by owners. This holds t rue regardless of
t he degree of complement arity bet ween pat ent s in t he company’s product ion
process.
5.2
D et err ence
The analysis t ells us t hat in t he absence of t ight wealt h const raint s, t here
will be ine¢ cient delay. This suggest s an int riguing possibilit y. Suppose
t he pharmaceut ical company faces a …xed cost of ent ering bargaining. T he
delay problem, and t he fact t hat t he part ies divide and conquer, could be
su¢ cient ly severe t hat it is not wort hwhile for t he company t o pursue t he
purchase of pat ent s. This will happen whenever t he company’s expect ed
payo¤ falls below t he cost of ent ering int o t he bargaining process.
5.3
I nfor m at ion G at her ing and R enegot iat ion
Suppose now t hat each player can …nd out whet her t he ot her player is present
at t he bargaining t able at any given t ime. Wit h t his knowledge, a player can
decide whet her it wishes t o commence bargaining or wait unt il lat er t o do so.
I n part icular, not e t hat t his is only relevant at end of dat e n: Eit her player
can avoid making a period n agreement , aft er observing whet her t he ot her
is present , and wait unt il period l.
First consider t he equilibrium (p1; p2) = ( 1; 0) in t he previous model. (I n
t his equilibrium owner 1 chooses t o go t o t he bargaining t able at dat e n
when owner 2 chooses t o go t o t he t able at dat e l and vice versa.) Will
t his cont inue t o be an equilibrium when informat ion gat hering is possible?
Consider player 1’s decision when he arrives at t hebargaining t able, and …nds
t hat player 2 has made t he decision not t o show up. Will player 1 decide
occur, for t he same reason t hat (1,0) is an equilibrium in t he previous model.
By a symmet ric argument (0,1) will also cont inue t o be an equilibrium. Now
consider t he mixed st rat egy ( ¹p1; ¹p2). It is t rivial t o show t hat t here is no
incent ive t o deviat e from t his st rat egy. Suppose t hat bot h players arrive
at t he bargaining t able at dat e n. They choose t he same probabilit y of
exit ing, and bargaining at dat e l, for precisely t he same reason ( ¹p1; ¹p2) was
an equilibrium of t he previous game.
A st andard device t o eliminat e ine¢ ciencies in bargaining is t o int roduce
cost less renegot iat ion. I n t his model, t here is no incent ive for part ies t o
renegot iat e. Once t he company has purchased a pat ent , t he prior owner is
no longer st rat egically relevant .
6
C onclusion
We examine t he problem faced by a company t hat want s t o purchase t wo
complement ary pat ent s from dist inct owners. Our model capt ures t heprocess
by which t hese complement ary pat ent s areacquired and shows t hat ine¢ cient
delay can occur as a result of pat ent owners being st rat egic. While t he
ownership lit erat ure assert s t hat complement ary pat ent s should be owned
t oget her , we show t hat it is precisely t his sit uat ion t hat leads t o ine¢ cient
delay. I ndeed an increase in t he degree of complement arit y (via an increase
in v) will ult imat ely lead t o a higher probabilit y of delay. However when
t he probabilit y of delay is low, an increase in complement arit y leads t o a
reduct ion in t he chance of delay.
As well as changing t he degree of complement arity, we show t hat delay
decreases as t he discount fact or increases. Ext ensions include t he int r
oduc-t ion of wealoduc-t h consoduc-t rainoduc-t s, infor maoduc-t ion gaoduc-t hering and renegooduc-t iaoduc-t ion. I n all
delay can be eliminat ed when t he company has su¢ cient ly low wealt h.
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