On the design of bilateral contracts

ON

THE

DESICY

OF

BILATERAL

Seppo HONKAPOH

COWR,\C”J-S IA*

The plan 4 t.k paper i! ris follows. In section Z iile basic framcwrlrh I.. NT of thz marginal equal treatment prrbpcrty ;IWI ItI+ out. the cquivalencc Green Honkrpohja (I’ &3) class of contracts is proud. and the tlcsrpn ~W&IVI for optimal cJctfaCtS is formulated. In section 3 1 anaiyx Ihc dc~pr, w. The exists ncc of a solution is established, and a con~3trl~ctl~. method for dctem lning It is given. Stxtion 4 is devoicd to t hc c\; ,~nl + r-h IN &VI ,ing the surplus from trading. in section 5 the Irnpilc;lrl~~l ,*I 0% c\ Fc‘st the d~tcd

* nprLlach are dkusscd.

Irwrtturc

cn hargaining

Sonic

3:cv-icluG~w\

over contra

ts arc ofkrcd

ilftd

rcfl:rcncl*y $,I

in wtlon

1’1

Incentive

compatibility

dtites,

m&r suftibcnt d&mutUtmlaty.’ that 2.c

first order conditions

hold as identities in c and 3, see Gnxn and Honkrphja details. By ditierentiation and after a kw manrpubt~~s fUdW?ltGl

PqrtUti#l

((a + b)q + (C- &k&.4 + (a + bl9,9, =o.

for the one O~UJIIS rk (19M

Asi0m

1.

(Incicmenta!

equal treatment.)

margil, 41 ut111t) \h,lt, In wtbrds the axiom st,rtcv that if, say, the s&A one unit. thtn the contract ..houl+ be >-.;h that ils t’ff~ct (WI the* W: ,II, rk

hr:ver is the same as the effete 1~1IL

same shift II htrycr’.

11~11 11

,

.

-

’

!,

selkr’s net utility.’ Therefore. Axiom i implies that the: L’I ~W+I n1.t *\III k 1 the buyer dnd the seller dqknnd syrrmctricall;y 0;1 th; rca~~~d :~IUC\ rtf 11,. random prameters. No;e aLkl that Axiom 1 makes ‘WU\L’ tly III cdm)ljn ~~~~~~ with mcentivc compatibilAy. 3s otherwise it cannot be ve~rfi~~l I hc I h tr 1 terl7tit~~~n result mentioned is no* stated as

u hwh lmphcz that the L>quation

Next 1 define the set of admissibk cc rt~ In view of previous MS the analysis of optimal contracts is best ~~nductcd in terms of tbc quantity traded, so that by adopting Axiom 1. Proposition 1 implies that the set of admissible contracts is taken to bc

The next step in the analysis is lo introduce l& oplundily cnttfwn to lhc model. .‘h,x the framework iovdvcs ut&ties in mrry tcn~ w that utility is tranokrabk, it would sbcm rpprofwiate lo lake lbc s4uu cd utdilie Tbc d impwtubt oocLsjdcT1lRm as the 2pproprirtc w&UC fuIaha* stems from tbe ptesencz of uncertainty and private ~tkforuut.A m t& mockl. In the Mcraturc the so-called cx ante. interim, and cx post ~PQrorrcba have been proposed, 9cc e.g., Hammond ( 1981). rnd ~~olmslrhn and Mycrson wkn rk sum of a@mts’ (1983) which cantain many further m. thru approaches arc at! utitities is taken to be the wetfare function tbcs equivalent, as is easily checked. With these considerations in mind I define fkflnirion.

A contract is oprimol. if it s&m

In other words, optimal contracts arc three whwh mrrlmllr Ik CR~CXM total utility subject to being incentive wmpatrhlc rti pwscswnp ~hc prc~p~rt! of increment31 qua1 lrcatmenl. d It should be pointal out that 1 have not tntro&cwd any coasulmt~ns individual rationality as constraints in the &sign pro&m As will b m later, it is un ncocssary to do so at this stas. since the opimimtmon ~41 TV uarried out in terms of tk quantity lradai. One the quant~l) fuI)cthW ha* been determined the payment function can k wlvlrd h) lntefntm; the hrworder conditions (2.4) ;I..d (2.5). This ~iclcfs the funcrmr rt$.nl rhuh n umyur up to an additive constrnl. and il will turn WJ~ rh1 rndrvduai ralhm&lr s only relevant ior determining Ou value of the cnrrstant To knzgin rhc analysis of the optimlutmn notation

fmtkm

kl

UI mtr0duLx

It-+

TIcse tfansfomr the fundamental cpidinaty differential equation

eq. (2.6) into an autonomous PCCO~~I-~,T~:~~

?‘-‘?‘+(I -#=o. md the second-order dons1rain6

tZ I’DI conditions

and the non-negativity

of ~1n;lu

ImpI,

th,

lxt alao,l’(~I be the density function of the ranJon variable I -‘.I: o, ;in
Prtr$ The constraints are jtist the rewriting cf the requitemcnr th rt ye II To ~4c that maximizinp LtC; + t’) amounts to minimizing (2.12) one utilr/r\

the

wtd

obxrvrtion

I sm

is simply

rl t h < 0 compktcs

[( l/Z)(a + h) “]E~,.Ic~). a constant. QED. the proof.

and so the

Thcr&rt. In this model with quadratic utilities of the individu&. maalrntnngt the cx;rcutLTdvalue of the sum of utilities is equivalent to minimrAnK the cxpectcd r-alum of the squared deviations of the contracted quantity from tk

first-hcst solution. since J =(ta+h)(q-q+).

In rhc pcxding sectim the basic model of bilateral contracts was =I out, ad the w d optimal contracts was formulated in precise terms. In this

~tionIcoasiderthtoptimizrr~prabkn,iabtcrii.~~onIhcbrsisol Lemma 1 probkm (28) is in principk a varihmd prob&m. I prmxrd by solving the fundamental quation in the form (2 IO). It an IX shoa n [see (1983)]. that the solutions to (210) a obcrinr& in Grocn-iionkapohja parameter& form.

For i: proof see Green and Honkapohp (I’WLI. Several comnrcnts are in order. First, rhc rckvant

(rilrrrnctcr

d~nu~n

ts

Y A

/

Positive

branch

f-‘ig. I

in fig. I the graph of the general solution (3.4) ic; iiIustrittc4I I I-W t:‘rl~~ t-ranch‘ refers to the part of the solution in whlLh /i, f\ KIC\.I~ and In ~hc ‘pwitivc branch’ pz is applicable. Fnurth. tc- ensure that the function J*=J(.x-I defined hy 13.41 I\I~ 1-1 1111 cn~wc Jom;rm [ !, .i-j one simply stipulates the inequalities

‘rqativt

is obtainal from d#fcmMihng Ibe expression for since aUlag, 1 x(y pi) in (3.4j.T5xJ3,)
solution

r = rf r. 4 pum

hg tk

fint part r)f

the lemma and formula (3.4) satisfies

Next we give tk main theorem w&h pro\lde a c nstruct~~ solve for optima! contracts ?? rtd kads to the c\rsten~~c result

rncth~!

to

a single variable ZE [s,_f]. Note also that in (3.8) we have the V~IIC\ II I~ Using these and integration by parts we ob!-iin

-?‘ j F[(! 0

-I)(u+

lie

“+z](~-a~2u(u--1)e

4

‘“du

()I

1)

frc*$ The objective fun&n in (3.6) is evidently a continutru\ functI(prl .*I z sinclc F has a density f = F’. Sixe ZE is, X] by Weierstrass’ thcotcm J 1 81 attains its minimum at some value x,, which together with the valur ,hrr /I, and p:. given in Lemma 3, define the optimal contract quar tlty ~rlrlct~or~111 the pwametric form (3.4). Once the function q:Pi-+R+ hau IX :n found In thl, way. the payment function can be determined by intcgf atinf (2.4) dnd 12 ;I (Note that by crxnstruction the inteprabilicy con5:ior-r is sstisfxd.) 01 I) In general the optimal contract &pends critically on the p’rchh~!~ t\ distrihutron of the random variable x=&--J. Though Thearem I gtrek ,I constructive way to cornyule the optimal solc;ion, given a distrihutlcpn furhztmn F(x). the prtiblern of minimizing (3.6i over z F [I 5. .i ] is falrlj ~urn~~mc rt ‘IIIS level of generality. For example, the uniqu:ness of the rrptamal ?,>1q+,(1t4n is not guaranteed. As a special case permitting an explicit rnJ IJnqUc: ;dution let us consider the situation in which x has a symmetrbc 111uagulat distribution. As is well known. this case arises when I; and o arc mdcpndtnt and both ha\-e a uniform distribution ever interv:11s of cqu;~l kngth t

UJmplc

&nut)

15

8 ha\

;r symmct-ical

triangular

distribution

over [.x,.x’]. i.e.. It\

4~1trsn2. (Err anteequal

division.) Thecontract

(q.fr satisfies III I.’

!I

t,r b * :I

In dt\Ang the pains frm trade me should ;1Iso take lnttr itti(lellnl conudcrations of indi\idlial rationality, since they arc important In Jctcr mmrnp the apenls’ willrngness to participate in the contract. In analogy with three Jiffcrcnt viewpints for c.Jaluating social welfare which wcrc p)rntc4 0uI In elan 2 c~nc rnily forn;ulate three different concepts of indlvlc!u,ll tattl~n.rht!. nomel> cx ante. interim and ex post. Each WC of them mit! !W rrl~~;rnt m \;brltlus concrctc ;lpp’ications. but m what f.:,llouy 1 UIII 1lnl\ ~CUJS tbn rhc c~~nsqucnc:s of cx i nlc and cx post individual rallonalltj In IIIIS wwon WC wnsider

Comput.ing tbe dcrivrti%u o#

obcmm

Using notation (2.9) thu kmds to zy-yy’-x=0

Finally. let us introduce the concept ot ex post individual urnsidcr its relationship TV ex post equal division oi surplus. .4 WWI 3.

P~NI)

(Ex post individual

rationality.)

The rc~ult follows if it can be shown thrit mtn[tI’

t)i(r;tj)~~~==(C:-tI(!;..!1

mlnI(I’+t)!(r..6)Fh’]=(t*+t)(;I+3,3).

.- 2).

rdtlc:n,llrr.

,cncj