MATHEMATICS IN CONTRACT THEORY (PRELIMINARY VERSION)

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1 MATHEMATICS IN CONTRACT THEORY (PRELIMINARY VERSION) HAO XING For general inroducion o conrac heory, refer o Bolon and Dewaripon, 25]. For mahemaical heory for coninuous-ime models, refer o Cvianić and Zhang, 23].. One-period model.. Problem formulaion. To undersand he flavor of quesions in conrac heory, we sar wih an one-period model in his secion. Consider an agen whose uiliy is modelled by an exponenial uiliy wih he risk aversion, i.e., U A (c) = e c. If hired a ime, his agen manages a projec, which produces an oupu a ime. The oupu is random, and i is modelled by a normal random variable X wih he mean α and he sandard deviaion σ, i.e., X N(α, σ). The mean α is deermined by he agen s effor. If he agen works hard, α is high, oherwise α is low. The randomness of he oupu is characerized by σ which canno be conrolled by he agen. The effor is cosly o he agen. The cos is given by a cos funcion g(α) which is he moneary cos o he agen if he exers effor α. We consider he following cos funcion g(α) = 2 α2. If hired, he agen will receive a lump-sum compensaion ξ a ime. We call his compensaion he conrac paymen, or simply conrac. The agen chooses his opimal sraegy o maximize he expeced effor ne cos, i.e., E U A (ξ 2 α2 ) ] Max. Assuming ha his opimizaion problem has an opimizer α ξ, EU A (ξ 2 (αξ ) 2 )] is he opimal value associaed o he conrac ξ for he agen. The agen has a reservaion value U A (R), which describes he minimal uiliy ha he agen needs o ake on a job; i.e., he agen will only ake on a conrac if he associaed opimal value is no smaller han his reservaion value. Consider a principal whose uiliy is modelled by an exponenial uiliy wih he risk aversion, i.e., U P (c) = e c. The principal wans o hire he agen o work on he projec. However, he principal canno monior or observe agen s effor. The principal can only observe he realizaion of X a ime. (I is impossible o esimae he mean Dae: June 24, 27.

2 MATHEMATICS IN CONTRACT THEORY 2 of a normal random variable wih only one realizaion.) Therefore, here is informaion asymmery beween he agen and he principal. This informaion asymmery is called moral hazard in he economics lieraure. A ime, he principal receives he oupu X and pays he agen ξ. The principal aims o choose conrac ξ o maximize her expeced uiliy, i.e., E U P (X ξ ξ) ] Max, subjec o E U A (ξ 2 (αξ ) 2 ) ] U A (R). Here X ξ is he oupu when he agen employs he effor α ξ. The consrain is called agen s paricipaion consrain. The goal of his problem is o find principal s opimal conrac ξ and agen s opimal effor a ξ..2. Agen s opimizaion problem. Firs, i is no opimal for he principal o pay a consan salary ξ, because if agen receives a compensaion which does no depend on he oupu, he will no exer any cosly effor a all. depends on X. Le us consider he following linear conrac Therefore, ξ needs o be random and ξ = ax + b, a, b R, (.) where b is he fixed salary and ax is he performance-based compensaion. The consans a and b are called conrac sensiiviies. For a given linear conrac in (.), he agen wans o maximize he expeced uiliy Recall ha The previous expeced uiliy equals o E e (ax+b 2 α2 ) ]. Ee cx ] = e αc+ 2 σ2 c 2. (.2) e b aα+ 2 σ2 a 2 γ 2 A + 2 α 2. Is opimizer minimizes aα + 2 α 2. Therefore, he opimal effor is and agen s opimal value is a ξ = a, (.3) e b+ 2 σ2 a 2 γ 2 A 2 a 2.

3 MATHEMATICS IN CONTRACT THEORY 3.3. Principal s opimizaion problem. Given agen s opimal effor a ξ and he associaed oupu X ξ, he principal aims o maximize E U P (X ξ ξ) ] = E e (X ξ ax ξ b) ] = e b ( a)a+ 2 σ2 γ 2 P ( a)2, where (.2) is used o obain he second ideniy. (.4) Observe ha principal s opimal uiliy is decreasing in b. Therefore he principal wans o choose he smalles b such ha he paricipaion consrain is saisfied. This implies leading o b = 2 σ 2 a 2 2 a2 + R. e b aα+ 2 σ2 a 2 γ 2 A + 2 α 2 = e R, Plugging he previous expression of b back o (.4), we obain E U P (X ξ ξ) ] = exp ( R + 2 σ 2 a 2 2 a 2 ( a)a + 2 σ2 γ 2 P ( a) 2). Therefore, we jus need o minimize he exponen of he righ-hand side. The opimizer is a = + σ 2 + ( + )σ 2. Proposiion.. Among all linear conracs of ype (.), he opimal linear conrac is given by ξ = a X + b, where a = + σ 2 + ( + )σ 2 and b = 2 σ 2 (a ) 2 2 (a ) 2 + R. Remark.2. From he expression of a, we can see ha a, when eiher σ,, or. Therefore, he principal sells he whole projec o he agen for cash a ime, when eiher he projec is riskless, he agen is risk neural, or he principal is exremely risk averse. Remark.3. The opimal conrac in he linear class is obained in he previous proposiion. However, if he principal is allowed o choose from nonlinear conracs, i is no clear wheher nonlinear conracs will improve her uiliy. Consider a N-period model, where he oupu a he end of period N is X + X 2 + X N. If he principal can observe all X i, she can choose from pah-dependen conracs f(x, X 2,..., X N ). In his case, i is resricive o consider only linear conracs. We will ackle hese problems in a coninuous-ime framework in he nex secion. 2.. Preliminaries. 2. Holmsröm and Milgrom (987)

4 MATHEMATICS IN CONTRACT THEORY Problem formulaion. We presen Holmsrom and Milgrom, 987] in his secion. Le (Ω, F, (F,T ] ), P) be a filered probabiliy space, where he filraion is he augmened filraion generaed by a -dimensional sandard Brownian moion B. The agen, if hired a ime, works for a projec coninuously from ime o T. His effor is an adaped process α. The oupu process X saisfies he dynamics where σ is a posiive consan, dx = α d + σdb α, (2.) B α = B α s σ ds. (2.2) One can observe ha X = σb. Define a new probabiliy measure P α via dp α dp = exp ( α s σ db s 2 ( αs σ )2 ds ). We assume ha α saisfies enough inegrabiliy so ha he righ-hand side has expecaion under P. Therefore, P α is a probabiliy measure equivalen o P. In wha follows, we denoe E Pα ] by E α and E Pα F ] by E α ]. By Girsanov heorem, B α is a Brownian moion under P α. Therefore, if α is deerminisic, X T has normal disribuion under P α wih he mean α d and he sandard deviaion σ T. Remark 2.. The formulaion in (2.) is called he weak formulaion. In such a formulaion, conrolling he drif of X is equivalen o conrolling he probabiliy measure P α. The cos of agen s effor is assumed o be sraegy o maximize E U A (ξ 2 α2 sds. The agen chooses his opimal 2 α2 sds) ]. Assume ha his problem admis an opimal sraegy α. The principal observes he oupu process coninuously. However, she canno disinguish he drif α from he noise. In oher words, when he oupu process increases, she does no know if he agen works hard or she is jus lucky. This informaion asymmery beween he principal and agen creaes moral hazard. Given agen s opimal effor a ξ and he associaed oupu process X ξ, he principal aims o maximize subjec o agen s paricipaion consrain. E U P (X ξ T ξ)],

5 MATHEMATICS IN CONTRACT THEORY Agen s opimizaion problem. Le us firs define a dynamic version of agen s value funcion: v := ess sup α E α UA (ξ 2 α2 sds) ]. (2.3) One can hink v as he opimal value a ime if he agen acs opimally from ime onwards. Therefore v is called he coninuaion value. Define u via U A (u ) = v. (2.4) The process u is called he cerainy equivalence, which is he moneary equivalence of agen s opimal value. Now we presen a heurisic argumen o idenify he dynamics of u. Assume ha u has he following dynamics where H will be deermined in wha follows. Since U A is an exponenial uiliy, γ e u e du = H d + Z db, 2 α2sds = ess sup α E α exp ( ( T ξ 2 α2 sds ))]. The maringale principal expecs ha γ e u e γ A 2 α2 s ds is a supermaringale under P α for any sraegy α; γ e u e γ A 2 α2sds is a maringale under P α for he opimal sraegy α. We will use his maringale principal o idenify H, hence he dynamics of u. To his end, applying Iô s formula, we obain d γ e u e e u e 2 α2 sds = H + γ 2 α2 s ds 2 AZ α2 ] d Z db = H + 2 Z α2 Z α σ where he second ideniy follows from (2.2). Having γ e u e 2 α2 s ds as a supermaringale implies ha H + 2 Z α2 Z α σ Moreover, having one α such ha γ e u e previous inequaliy is an ideniy when α = α. Therefore, d Ω a.s. ] d Z db α, 2 (α s )2 ds is a maringale implies ha he and he opimal α is H = inf α { 2 α2 Z α σ = 2 ( σ 2 )Z 2, α = Z σ. } + 2 Z 2

6 Therefore, he maringale principal implies ha MATHEMATICS IN CONTRACT THEORY 6 du = 2 ( σ 2 )Z 2 d + Z db, u T = ξ, (2.5) where he erminal condiion follows from (2.3) and (2.4). One can view (2.5) as a Backward Sochasic Differenial Equaion (BSDE), whose soluion is a pair of adaped processes (u, Z). The BSDE (2.5) can be solved explicily. Consider he process e ( σ 2 )u. Iô s formula implies ha de ( σ 2 )u = ( σ 2 ) e ( σ 2 )u Z db, e ( σ 2 )u T = e ( σ 2 )ξ. Given ha E e ( σ 2 )ξ ] <, we have e ( σ 2 )u = E e ( σ 2 )ξ ], and Z is idenified by he maringale represenaion heorem. In conclusion, for any conrac ξ such ha E e ( σ 2 )ξ ] <, we can idenify (u, Z) saisfying (2.5). This pair is also unique in he class where e ( σ 2 )u is of class (D). The equaion (2.5) also provides a handy represenaion of he conrac ξ. Given u and he process Z, running he dynamics (2.5) forward in ime yields ξ = u + 2 A )Z 2 σ 2 d + Z db. (2.6) This represenaion will be useful for principal s opimizaion problem. We have derived he dynamics of u from he maringale principal. Le us now sar from (2.5) and verify ha is soluion is he cerainy equivalence defined in (2.4). Proposiion 2.2. We resric he conrac ξ and agen s sraegy α o he class such ha he BSDE (2.5) admis a soluion (u, Z) and exp ( (u 2 α2 sds) ) is of class (D). Then agen s opimal sraegy is α = Z/σ and u is agen s cerainy equivalence. Proof. For arbirary α, he consrucion of H ensures ha e γ A(u 2 α2 s ds) is a local supermaringale. Since we assume ha his process if of class (D), i is also a supermaringale. Therefore U A (u ) = e u E α e ( u T 2 α2 sds) ] = E α ( e ξ ]. 2 sds) α2 When α = α, e γ A(u 2 (α ) 2 sds) is a local maringale, hence a maringale due o is class (D) propery. Therefore, he previous inequaliy is an ideniy, verifying he opimaliy of α.

7 MATHEMATICS IN CONTRACT THEORY Principal s opimizaion problem. Using (2.6), principal s expeced uiliy is reduced o E α UP (X ξ T ξ)] = E α ( ( exp = E α ( ( exp u + Z d + σ σdb α u ( Z σ 2 ( + σ 2 )Z 2 where (2.2) is used o obain he second ideniy. sochasic exponenial E ( We obain (σ Z )db α ) = exp ( ))] (γ 2 A )Z 2 σ 2 d Z db ) d + (σ Z )db α ))], Rewrie he righ-hand side using he (σ Z )db α 2 γ2 P ) (σ Z ) 2 d. where E α UP (X ξ T ξ)] ( = E exp u + ) ( f(z )d E )] (σ Z )db α, f(z) = 2 γ2 P (σ Z) 2 Z + γ ( ) σ 2 P γa + σ Z 2. 2 (2.7) Se Z = argminf(z). Calculaion yields Z + σ 2 = σ + σ 2 ( + ). Now we are ready o presen he opimal conrac for he principal. In conras o he one-period model, a linear conrac is proved o be opimal in a large class of conracs, which saisfies proper inegrabiliy condiions and includes nonlinear and pah-dependen conracs. Theorem 2.3. Consider he class of conracs specified in Proposiion 2.2, moreover E ( γ p (σ Z )db Z/σ) is a P Z/σ -maringale. Then he linear conrac ξ = + σ 2 + σ 2 ( + ) X T + b, where b = R + 2 ( σ 2 )(Z ) 2 T, is he opimal conrac. Proof. From (2.7), we can see ha principal s expeced uiliy is a decreasing funcion of u. On he oher hand, u is agen s cerainy equivalence a ime. Therefore, in order o

8 MATHEMATICS IN CONTRACT THEORY 8 saisfy he paricipaion consrain, he principal chooses u = R. From (2.7), we also obain E α UP (X ξ T ξ)] e R+f(Z )T E α ( E = e R+f(Z )T, )] (σ Z )db α where he ideniy follows he assumpion ha he expecaion of he sochasic exponenial is. The inequaliy above is an ideniy, when Z in (2.6) is chosen as Z. Recalling ha B = X/σ, we have verified he opimaliy of he conrac in he saemen. 3. Sannikov (28) 3.. Problem formulaion. Raher han a conrac of lump sum paymens in he las secion, we consider a model of coninuous paymens in Sannikov, 28]. The oupu process X is formulaed weakly as in (2.), whose drif is conrolled by he agen. The agen s preference is represened by a uiliy funcion U and he rae of cos for agen s effor is given by a funcion g. We do no specify explicily forms for U and g, only assume ha g is sricly convex. Raher han he moneary cos as in he las secion, we consider g(α) as he cos o agen s uiliy per uni of ime. A conrac is a sream of paymens {c ; }. The agen receives he cash flow c and consume i righ away wihou saving. The agen s opimizaion problem is re α e r( u(c ) g(α ) ) d ] Max, where r is he discouning rae and he r in fron of he expecaion is a normalizaion consan. We assume ha agen s opimizaion problem admis an opimal sraegy α. Agen s reservaion uiliy is R; i.e., he agen will no work for he projec if he opimal value is less han R. Define agen s coninuaion value as W = ess sup α re α e r(s )( u(c s ) g(α s ) ) ds ]. (3.) The agen has limied liabiliy. He does no accep negaive W in any siuaion. The principal is risk neural. She chooses he compensaion sream c o maximize her expeced profi sup re α τ τ e e r( ) ] α c d, c,τ e where τ is he firs ime ha agen s coninuaion value reaches zero, i.e., τ = inf{ : W = }, and τ e. A ime τ, he conrac is erminaed, he agen is no longer paid and he exers zero effor afer τ. A ime τ e, he principal reires he agen by paying him a

9 MATHEMATICS IN CONTRACT THEORY 9 consan rae forever, hen agen sops working. This consan reiremen rae is deermined so ha agen s coninuaion value remains he same a reiremen ime Agen s opimizaion problem. Le us firs use he maringale principle o heurisically derive he dynamics of agen s coninuaion value W. Define W as W = e r W + r e rs( U(c s ) g(α s ) ) ds. Given c, we expec from he maringale principle ha W is a supermaringale under P α for arbirary sraegy α, and a maringale under P α he dynamics Iô s formula implies ha dw = H d + rσz db. for he opimal α. Suppose ha W has d W =e r( H d + rσz db rw d + r ( U(c ) g(α ) ) d ) =re r( r H W + α Z + U(c ) g(α ) ) d + re r σz db α. The maringale principle implies ha he drif is nonnegaive for arbirary α and zero for he opimal α. Therefore, H = r ( inf α {g(α) αz} + W U(c)) = r ( ĝ(z) + W U(c) ), where ĝ(z) = inf α {g(α) αz} and he opimal sraegy α = (g ) (Z). As a resul, W follows he dynamics dw = r ( ĝ(z ) + W U(c ) ) d + rσz db. (3.2) Raher han a erminal condiion a a finie ime, he erminal condiion for (3.2) is replaced by he following ransversaliy condiion lim Eα ] e rτ W τ =, for sraegy α and any sopping ime τ. (3.3) τ From a BSDE poin of view, (3.2), ogeher wih (3.3), is called an infinie horizon BSDE, sudied by Royer (24). The following resul makes he heurisic argumen rigorous. Le us resric he compensaion sream and agen s sraegy α such ha E α e r U(c )d ] < and E α e r g(α )d ] <. Proposiion 3.. For a given compensaion sream c, suppose ha here exiss processes W and Z saisfying (3.2) and (3.3). Then α = (g ) (Z) is agen s opimal sraegy and W is agen s coninuaion value defined in (3.).

10 MATHEMATICS IN CONTRACT THEORY Proof. From he consrucion of H, we have ha W is a local supermaringale. localizaion sequence {τ n } of is local maringale par. We have Take a τn W E α e rτ n W τn + r e r( U(c ) g(α ) ) d ]. (3.4) Sending τ n on he righ-hand side, and using (3.3) and he dominaed convergence heorem, we obain W E α e r( U(c ) g(α ) ) d ]. For he sraegy α, he inequaliy in (3.4) is an ideniy. A similar limi argumen yields W = E α e r( U(c ) g(α ) ) d ], confirming he opimaliy of α Principal s opimizaion problem. The key observaion by Sannikov is ha agen s coninuaion value can be used as he sae variable for principal s problem. Consider he dynamics (3.2), where boh Z and c are considered as he conrol variables for he principal. Define principal s value funcion as τ τ F (W ) = sup E α r e r( ) α c d W = W ], c,z,τ where α = α(z) = (g ) (Z) and τ = inf{ : W = }. This is an opimal conrolsopping problem. To derive an equaion saisfied by F, we use he maringale principle again. To his end, consider and define F (W ) = ess sup c,z,τ E α(z) r M = r e r(s )( α(z s ) c s ) ds ], e rs( α(z s ) c s ) ds + e r F (W ). We expec from he maringale principle ha M is a supermaringale under P α(z) for arbirary c, Z, τ, and M is a maringale under P α(z) unil τ for opimal c, Z, τ. Assuming ha F C 2 (R), applying Iô s formula o M, we obain dm =e r r(α(z ) c ) rf (W ) + r ( ĝ(z ) + W U(c ) ) F (W ) + r2 σ 2 2 Z2 F (W ) ] d + re r σz F (W )db =re r α(z ) c F (W ) + ( g(α(z )) + W U(c ) ) F (W ) + rσ2 2 Z2 F (W ) ] d + re r σz F (W )db α(z).

11 MATHEMATICS IN CONTRACT THEORY The drif mus be nonnegaive for all conrols and zero for opimal conrols. Therefore, we obain he following Hamilon-Jacobi-Bellman (HJB) equaion for F : { = sup α(z) c F (W ) + ( g(α(z)) + W U(c) ) } F (W ) + rσ2 2 Z2 F (W ). (3.5) c,z This HJB equaion needs wo boundary condiions. On he one hand, F () =. (3.6) On he oher hand, define a reiremen funcion F (U(c)) = c. This is he value o he principal if she pays he agen a consan rae c. In his case, he agen no longer works. The oher boundary condiion is F ( W ) = F ( W ) and F ( W ) = F ( W ). (3.7) This is a free-boundary condiion, since W is deermined ogeher wih he soluion F. The condiion on he firs order derivaive is called he smooh pasing condiion. Now we are ready o sae he main heorem. Theorem 3.2. Suppose ha he HJB equaion (3.5) ogeher wih is boundary condiions (3.6) and (3.7) admi a soluion F C 2 (, W ) and W R, ). Le Z(W ) maximizes α(z) + g(α(z))f (W ) + rσ2 2 Z2 F (W ), and c(w ) maximizes c U(c)F (W ). Choose W = max W R, W ] F (W ) and define W via he following SDE dw = r ( ĝ(z(w ) + W U(c(W ))) ) d + rσz(w )db, W = W. Then principal s opimal conrac is given by c(w ), agen s opimal effor is α(z(w )), τ = inf{ : W = }, and τ e = inf{ : W = W }. Remark 3.3. The proof is a verificaion argumen, see Sannikov, 28] and Srulovici and Szydlowski, 25 In paricular, in order o ensure (3.5) admis a C 2 soluion, one needs o resric Z Z for some posiive consan Z. This ensures (3.5) is uniformly ellipic, bu resrics principal s admissible conracs. An numeric example is given in Figure. 4. DeMarzo and Sannikov (26) 4.. Problem formulaion. We presen DeMarzo and Sannikov, 26] in his secion. The agen, if hired, produces an oupu process X which follows dx = (µ α )d + db α,

12 MATHEMATICS IN CONTRACT THEORY 2 where µ is a consan and db α = db + α d. The agen s effor α is a shirking acion, which can be inerpreed as inefficien implemenaion or working less hard han he should be. This shirking acion yields privae benefi o he agen. The agen can choose any nonnegaive α. The benefi is described by a parameer λ, ]. The larger λ is, he higher benefi agen ges. The cumulaive compensaion paid by he principal is described by a non-decreasing process C. The agen is risk neural and has a reservaion uiliy R. Given C, Agen s opimizaion problem is E α e γ( dc + λα d )] Max. Le α be agen s opimal sraegy for he problem above. The principal pays he agen a non-decreasing process C. The agen has limied liabiliy. When his coninuaion value reaches zero, he projec is liquidaed, he principal ges L and he agen ges nohing more and sops shirking. Principal s opimizaion problem is subjec o agen s paricipaion consrain. τ E α e r (µ α )d dc ] + e rτ L ], 4.2. Agen s opimizaion problem. Define agen s coninuaion value W = ess sup α E α e γ(s )( dc s + λα s ds )]. Using he maringale principal similarly as before, we obain he dynamics of W as dw = γw + inf α {α (Z λ)} ] d dc + Z db. (4.) When Z λ, he opimizer α = and inf α {α (Z λ)} =. If Z < λ has posiive measure under d dp, he drif of (4.) is. Therefore, wellposedness for agen s problem impose a consrain for he conrac sensiiviy We also need he ransversaliy condiion Z λ d dp a.s. (4.2) lim Eα ] e rτ W τ =, for sraegy α and any sopping ime τ. (4.3) τ We consrain he compensaion process C and agen s sraegy α o he classes which saisfy he following inegrabiliy propery. E α e r U(c )d ] < and E α e r g(α )d ] <. An argumen similar o Proposiion 3. yields

13 MATHEMATICS IN CONTRACT THEORY 3 Proposiion 4.. Le W and Z be wo processes saisfying dw = γw d dc + Z db, he consrain (4.2) and he ransversaliy condiion (4.3). Then α = is agen s opimal sraegy and W is agen s coninuaion value Principal s opimizaion problem. Consider W as he sae variable for principal s opimizaion problem. Define principal s value funcion as τ F (W ) = ess sup Z λ E e r(s ) µds dc s ] + e r(τ ) L ], where τ = inf{s : W s }. This is called a singular conrol problem. Applying he maringale principal o principal s opimizaion problem, assuming ha C is differeniable, we can ge he following HJB equaion rf (W ) = max Z λ,c { µ + γw F (W ) ( + F (W ))C + Z2 2 F (W ) }. (4.4) However, C can be arbirarily large, in order o keep he HJB equaion well-posed, we need F (W ). This inequaliy can also be undersood from an economics poin of view. The principal can always pay he agen C >. In his case, agen s coninuaion value decreases by C F (W ) F (W C) C, which implies F (W ). On can also see from (4.4) ha C = whenever F (W ) >. Therefore, he principal delay he paymen o he agen unil F (W ) =. The boundary condiions are F () = L, F ( W ) =, and F ( W ) =. The hird condiion is he smooh pasing condiion. The boundary condiions a W imply rf ( W ) + γ W = µ. Tha is, he paymens are posponed unil he projec s expeced reurn is used up by he sum of he individual expeced reurns. Theorem 4.2. Suppose ha γ > r. Consider he ODE sysem : µ + γw F (W ) + 2 λ2 F (W ) rf (W ) =, F (W ), W, W ), F (W ) = W W, F () = L, F ( W ) =. Assume ha i admis a concave soluion F C 2 and W <. Then,

14 (i) F is Principal s value funcion. MATHEMATICS IN CONTRACT THEORY 4 (ii) When W, W ], ruh-elling is opimal, i.e., a. Moreover, i is opimal o se Z λ, and he paymens C o be he reflecion process which keeps W wihin, W ]. Tha is, C is he smalles increasing process such ha W = W + γw s ds C + λb says wihin, W ]. In paricular, when W (, W ), dc =. The conrac erminaes once W his. (iii) When W > W, hen he opimal conrac pays an immediae paymen of W W o he agen, and he conrac coninues wih he agen s new iniial uiliy W. Proof. Le F denoe he soluion o he ODE sysem and ˆF denoe he value funcion. We firs show ha ˆF F. To see ha, inroduce G := C + e rs (µ α s )ds dc s ] + e r F (W ). By Iô s formula we have ] e r dg = µ α + Z F (W )] + γw F (W ) + 2 Z2 F (W ) rf (W ) d + F (W )]dc + Z F (W )db α. Since F (W ) for W, W ], we have + lf (W ), + F (W ). This, ogeher wih he assumpions ha F is concave and Z λ, implies ha e r dg µ + γw F (W ) + ] 2 λ2 F (W ) rf (W ) d + Z F (W )db α. When W, W ], he drif is zero. When W > W, by rf ( W ) + γ W = µ and oher boundary condiions, we can compue ha he drif is r γ]w W ] <, hanks o r < γ. Therefore, in boh he cases we have e r dg Z F (W )db α. Thus, G is a P α -supermaringale. Noice ha F (W ) = G and F (W τ ) = F () = L. We have hen F (W ) = G E α G τ ]. for all a, τ. Taking he supremum, we ge F = G ˆF. For he sraegy in he heorem, G is a maringale. e r dg = Z F (W )db.

15 MATHEMATICS IN CONTRACT THEORY 5 Then, we have F (W ) = G = EG τ ]. Thus, he upper bound is aained, and ˆF (W ) = F (W ) for W, W ]. Finally, for W > W, noe ha, by he immediae paymen argumen above, he value funcion saisfies, for W > W > W, ˆF (W ) ˆF ( W ) W + W Since we have ˆF ( W ) = F ( W ) = F (W ) W + W, we have ˆF (W ) F (W ) Implemenaion Using Sandard Securiies. We now wan o show ha he above conrac can be implemened using real-world securiies, namely, equiy, long-erm deb and credi line. The implemenaion will be accomplished using he following: - The firm sars wih iniial capial K and possibly an addiional amoun needed for iniial dividends or cash reserves. - The firm has access o a credi line up o a limi of C L. The ineres rae on he credi line balance is γ. The agen decides on borrowing money from he credi line and on repaymens o he credi line. If he limi C L is reached, he firm/projec is erminaed. - Shareholders receive dividends which are paid from cash reserves or he credi line, a he discreion of he agen. - The firm issues a long (infinie) erm deb wih coninuous coupons paying a rae x. If he firm canno pay a coupon paymen, he projec is erminaed. The agen will be paid by a fracion of dividends. We assume ha once he projec is erminaed he agen does no receive anyhing from his holdings of equiy. Here is he resul ha shows precisely how he opimal conrac is implemened. Theorem 4.3. Suppose ha he credi line has ineres rae γ, and ha he long-erm deb saisfies x = µ γc L. (4.5) Assume he dividends are paid only a he imes he credi line balance his zero, making he credi line balance he process ha reflecs a zero. If he agen is paid by a proporion λ of he firm s dividends, he will no misrepor he cash flows, and will use hem o pay he deb coupons and he credi line before issuing dividends. Denoing he curren balance of he credi line by M, he agen s expeced uiliy process saisfies W = λ(c L M ). (4.6) If in addiion C L = W /λ (4.7) hen he above capial srucure of he firm implemens he opimal conrac.

16 MATHEMATICS IN CONTRACT THEORY 6 Proof. Denoe by δ he cumulaive dividends process. By ha, we mean ha dδ is equal o whaever money is lef afer paying he ineres γm d on he credi line and he deb coupons xd. Since he oal amoun of available funds is equal o he balance of he credi line M plus he repored profi X, and since M +X is divided beween he credi line ineres paymens, deb coupon paymens and dividends, we have dm = γm d + xd + dδ dx Wih W as in (4.6), and from (4.5), we have dw = λdm = γw d λdδ + λdb If we se dc = λdδ, hen his corresponds o he agen s uiliy wih zero savings and Z λ, which implies ha he agen will no have incenives o misrepor. Moreover, since he dividends are paid when M =, which, by (4.7) is equivalen o W = W, we see ha he opimal sraegy is implemened by his capial srucure. References Bolon and Dewaripon, 25] Bolon, P. and Dewaripon, M. (25). Conrac Theory. MIT Press. Cvianić and Zhang, 23] Cvianić, J. and Zhang, J. (23). Conrac Theory in Coninuous-Time Models. Springer. DeMarzo and Sannikov, 26] DeMarzo, P. M. and Sannikov, Y. (26). Opimal securiy design and dynamic capial srucure in a coninuous-ime agency model. Journal of Finance, 6: Holmsrom and Milgrom, 987] Holmsrom, B. and Milgrom, P. (987). Aggregaion and lineariy in he provision of ineremporal incenives. Economerica, 55: Sannikov, 28] Sannikov, Y. (28). A coninuous-ime version of he principal-agen problem. Review of Economic Sudies, 75: Srulovici and Szydlowski, 25] Srulovici, B. and Szydlowski, M. (25). On he smoohness of value funcions and he exisence of opimal sraegies in diffusion models. Journal of Economic Theory, 59:6 55.

17 MATHEMATICS IN CONTRACT THEORY 7 Figure. Sannikov (28) DeMarzo and Sannikov (26)