Strategic Incentives for Innovations and Market Competition

Transcription

1 Strategc Incentves for Innovatons and Market Competton Evangela Chalot Yale Unversty Konstantnos Serfes y Drexel Unversty Abstract We combne agency theory wth product market competton to study the strategc propertes of rvals contract decsons for cost-reducng nnovatons and the relatonshp between rsk and ncentves. We show that a change n rsk may trgger asymmetrc responses of rvals n the same market: lower rsk may nduce some rms to strengthen, whle other rms to weaken the ncentves provded to ther agents. Ths result holds regardless of the mode of competton n the product market, Cournot or Bertrand, as long as the rvals R&D decsons are strategc substtutes. Our model generates new emprcal mplcatons and can provde an explanaton for the lack of strong emprcal support n the lterature for a negatve relatonshp between rsk and ncentves. It also has polcy mplcatons about the e ect of rsk on the ncentves to nnovate. Keywords: Rsk, Power of ncentves, Product market competton, Innovaton. JEL: D8, L13, O30 Yale Unversty, Department of Economcs, 8 Hllhouse Ave., Room 08, New Haven, CT, 06511, USA. Emal: evangela.chalot@yale.edu, Tel.: , y Drexel Unversty, School of Economcs, 30 Market Street GHall, Phladelpha, PA, 19104, USA. Emal: ks346@drexel.edu, Tel.:

2 1 1 Introducton We consder agency theory wth product market competton to address the queston of how pay-forperformance depends on rsk when prncpals ( rms) strategcally nteract n the stage where contracts are chosen. Do all prncpals adjust pay-for-performance qualtatvely the same way when rsk ncreases? If not, whch prncpals are more lkely to strengthen and whch prncpals are more lkely to weaken the ncentves when rsk ncreases? We argue that hgher rsk trggers asymmetrc responses of product market rvals: when the provded compensaton contracts are strategc substtutes, a negatve and a postve relatonshp between rsk and ncentves can coexst n the same market. The relatonshp between rsk and pay-for-performance has receved sgn cant attenton n contract theory. The conventonal wsdom n models wth moral hazard, orgnatng from Holmström (1979) and Holmström & Mlgrom (1987), s that the optmal contract balances an ncrease n rsk wth weaker ncentves for e ort due to rsk-sharng between the prncpal and the agent. Hgher rsk ncreases the value of provdng nsurance to the agent and thus ncentves optmally decrease. However, the emprcal support for ths predcton s mxed (Prendergast (00)). In partcular, emprcal analyss of exstng contractual arrangements n many uncertan envronments - e.g., executve payments n knowledge based ndustres and n nancal sector, sales or franchse agreements - have, n many cases, unveled a postve lnk between rsk and ncentves. 1 paper shows that the latter result can also hold n a theoretcal framework where rval managers, who have d erent exposure to rsk, compete n the product market. We consder two rsk-neutral rms that, pror to competton n the product market (Cournot or Bertrand), hre rsk-averse managers (the agents) to conduct cost-reducng nnovatons (R&D). The R&D outcome n each rm depends on ts manager s unobservable e ort and the realzaton of a project-spec c shock. The shocks that ht the rvals R&D productons are correlated. Ths Thus, the owner of each rm (the prncpal) o ers to ts agent a rsk-sharng contract (Holmström & Mlgrom (1987)) that spec es the payment as a functon of both rvals actual cost reductons. ;3 We allow for two knds of asymmetres whch are crucal for our results: ether (a) rms are exposed to the same amount of rsk (measured by the varance of a common shock) whle the agents have d erent degrees of rsk averson, or (b) agents wth same degrees of rsk averson are apponted by rms that are subject to d erent levels of rsk (measured by the varance of dosyncratc shocks). When the contract n the rval rm s xed, hgher rsk nduces the rm n queston to o er a contract wth lower pay-for-performance. Ths negatve relatonshp may cease to hold when we allow the rval rm to respond by adjustng ts contract as well untl we reach the new equlbrum. Weaker ncentves mply hgher 1 More recent theoretcal works on moral hazard attempted to generate a postve relatonshp by consderng, for nstance, the role of nput montorng or endogenous matchng - they are dscussed n the lterature revew later n ths secton. However, the rsk-ncentves relatonshp s always negatve n models based on Holmström (1979). Prendergast (1999) provdes a revew of the prncpal-agent lterature. 3 Cost-based schemes are consstent wth real-world contractng practces. For nstance, the German Employees Inventons Act passed n 1957 establshes by law that nventors compensaton schemes wll be based on the expected value of ther R&D-outputs.

3 margnal cost (n expectaton) and lower sales. But n an asymmetrc olgopoly envronment, sales need not decrease the same for all rval rms. Thus, n terms of market share, some rms may bene t. Indeed, the rm that has hred the less rsk averse manager or s exposed to relatvely lower level of rsk due to dosyncratc shocks can take advantage of the exstng opportuntes for busness expanson. In turn, the hgher market share nduces the owner of the latter rm to strengthen the ncentves provded to ts agent. Ths s the new e ect that we dentfy -.e., named busness stealng e ect - and works aganst the standard nsurance e ect. When the busness stealng e ect s strong enough, t can generate a postve relatonshp between rsk and ncentves n the rm that gans market share. Thus, the e ect of rsk on ncentves need not be the same even for rms n the same ndustry. It depends on the relatve cost of ncentvzng the agents across product market rvals. Competton between rms and asymmetrc responses to an ncrease n rsk are necessary condtons for ths result. In addton, ncentves must be strategc substtutes so that as ncreasng rsk dmnshes a rm s appette for nnovaton, resultng n a loss n busness n the product market, the wnner rm n terms of market share can provde stronger ncentves. In the lnear demand case, ncentves are strategc substtutes n both Bertrand and Cournot settngs: the nature of strategc nteractons n the R&D market does not depend on the mode of competton n the product market. However, when demand s nonlnear, addtonal condtons about the curvature of the demand need to hold. 4 We dscuss these condtons and show that as long as ncentves are strategc substtutes, the relatonshp between rsk and ncentves can turn out to be postve. 5 The e ects of the moral hazard problem on the equlbrum ncentves of product market compettors has been analyzed by Hart (1983), Hermaln (199), Schmdt (1997), Rath (003) and Pccolo, D Amato & Martna (008), among others. 6;7 In a smlar framework as ours but wth symmetrc rms, Chalot (015) argues that, due to product market competton, rvals exert such hgh levels of R&D that they burn up ther pro ts. 8 In the presence of moral hazard, underprovson of R&D ncentves due to rsk-sharng can generate consderable cost-savngs, mplyng hgher pro ts for both rvals. Thus, hgher rsk can make rms better o, because t exactly decreases agents equlbrum e ort. However, these papers assume symmetrc 4 Our results n ths dmenson are consstent wth the lterature that has examned the queston of whether statc strategc complementartes/substtutabltes translate nto dynamc ones n mult-stage games, Vves (009). Amr & Wooders (000) analyze a two-stage game wth R&D decsons and product market competton ether n prces or quanttes (but no moral hazard). In the lnear demand case, the R&D decsons are strategc substtutes. 5 When ncentves are strategc complements, the relatonshp between rsk and ncentves s always negatve for all rms. 6 Rath (003) examnes the e ect of competton on ncentves, when there are changes n the number of compettors, the market sze, the transportaton cost or the cost of entry. Nckell (1996) and Vckers (1995) revew the exstng works on the relatonshp between competton and ncentves, whle Vves (008) provdes a survey of the exstng lterature on the e ect of competton on nnovaton. Technology adopton ncentves of market rvals are analyzed n Mllou & Petraks (011). Amr, Amr & Jn (000) also examne rms R&D strategc nteractons when Cournot compettors nnovate smultaneously or sequentally n the presence of R&D spllovers. 7 Aghon, Bloom, Blundell, Gr th & Howtt (005), among others, study emprcally the e ect of competton on ncentves. Gr th (001) and Baggs & De Bettgnes (007) examne the relatonshp between competton and agency cost. 8 Chu, Eeckhoudt & Rey (01) study the behavor of the relatve and partal rsk-averson measures. Mrrlees & Ramondo (013) analyze strateges n a contnuous-tme prncpal-agent model.

4 3 or homogeneous rms and derve a postve relatonshp between rsk and pro ts (not ncentves). The exstng lterature that derves a postve relatonshp between rsk and pay-for-performance consders d erent settngs. Lafontane & Slade (001) argue that t s the stronger ncentves that cause a hgher pro t volatlty. So, the postve relatonshp s the outcome of a reverse causaton. Ackerberg & Bottcn (00), Serfes (005) and Serfes (008), among others, hghlght the mechansm of endogenous matchng between prncpals and agents. The latter models consder that prncpals compete for hgh qualty agents, but assume away any drect competton among contracts, once prncpals have matched wth agents. When the (stable) matchng s negatve assortatve, low rsk-averse agents match wth hgh rsk projects (prncpals). Low rsk averson mples that the agent can tolerate stronger ncentves, and so n equlbrum, they can derve a postve correlaton between rsk and ncentves. Wth endogenous matchng (but no drect competton among contracts), ths postve correlaton s observed only across all prncpal-agent pars (due to endogenous sortng). Prendergast (00) departs from the standard rsk-sharng model and hghlghts the role of montorng n contractual agreements. In rsky envronments, montorng of the agent s actons s more d cult. As a result, the prncpal gves more dscreton to the agent and the contract entals hgh powered ncentves. Rath (003) consders an endogenous number of symmetrc rms that compete n prces along a Salop crcle. When the degree of product substtutablty ncreases more rms wll enter the market. Ths has two e ects: ) ncentves decrease and ) the (endogenous) varance of pro ts decrease. Thus, although exogenous rsk and ncentves are stll negatvely related, there exsts a postve correlaton between the varance of pro ts and ncentves. The underlyng mechansm and the thrust of the results n exstng works are very d erent from ours. We focus on the rsk faced by agents and hghlght the role of competton among managers n shapng the contract characterstcs. Rsk s exogenous and a ects asymmetrcally the rvals equlbrum ncentves. We show that a postve relatonshp between rsk and ncentves can arse n a gven rm (prncpal-agent par). Thus, publc polces whose goal s to decrease market rsk n order to encourage rms to nnovate may have the opposte result. Frms wth more rsk-averse agents wll respond more heavly to a decrease n rsk, provdng stronger ncentves and nvestng more n R&D. However, rms wth less rsk-averse agents whose agency cost s lower wll have a medocre response. In fact, as ther rvals become more e cent by conductng cost-reducng nnovaton, they wll extend ther busness at the expense of the rms that have hred less rsk-averse agents. The latter rms that are also more e cent n exertng e ort, may nvest less as rsk decreases. Thus, we argue that polces amng n rsk reducton may unexpectedly nduce some rms n the ndustry to nnovate less, defeatng the purpose of such practces. One can verfy that our results wll contnue to hold n envronments that do not necessarly encompass product market competton - for nstance, n poltcal campagns - as long as actons n the rst stage are strategc substtutes (perhaps for other reasons), and the partes compete for market share. Beyond the obvous mportance of our analyss for the relatonshp between rsk and power of ncentves, t can also shed

5 4 new lght on other aspects of the rm s organzatonal structure, such as rm boundares. Aghon, Dewatrpont & Rey (004), Alonso, Dessen & Matouschek (008) and Hart & Holmström (010), among others, explore rm boundares and nternal organzaton. 9 In partcular, as rsk ncreases, the moral hazard rsk-sharng model predcts that the ncentves for vertcal ntegraton should also ncrease (Lafontane & Slade (007)). Ths s because hgh-powered ncentves that typcally exst outsde a rm become more costly and thus each rm wshes to rely less on the market. Based on our arguments, such a postve relatonshp between rsk and vertcal ntegraton need not hold. Frms that gan n market share as rsk ncreases and ther cost of provdng nsurance s small enough, can even utlze the ntra- rm con cts of nterests and operate under vertcal separaton. Therefore, we provde an alternatve explanaton for a negatve rsk-ntegraton relatonshp that has also found emprcal support (Lafontane & Slade (007)). The paper s organzed as follows. Secton presents the model. It dscusses the R&D technology and the compensaton contracts. Secton 3 solves the game and examnes the relatonshp between rsk and R&D ncentves when competton n the product market s Cournot. It rst performs the analyss when the market demand s lnear and then, t generalzes the results by consderng general demand and cost-of-e ort functons. In secton 4, we focus on the R&D motves of Bertrand rvals and dscuss whether the mode of competton n the product or R&D markets n uences the rvals responses to an ncrease n rsk to whch the agents are exposed. We summarze n Secton 5 where we also dscuss emprcal mplcatons that come out of our model. The model The market conssts of two pro t-seekng rms 1 and, ndexed by and j where 6= j. Each rm s run by a prncpal who s rsk-neutral. Each prncpal also apponts a rsk-averse researcher (manager/agent) whose cost-reducng R&D e orts are unobservable and non-contractble. The partes partcpate n a 3-stage game. In stage 1, each prncpal, smultaneously wth hs rval, o ers a contract that stpulates a pece-rate pay based on (observable) R&D outcomes n order to encourage hs agent to perform. In stage, f the agent accepts the o er, she chooses an e ort level. In stage 3, after the R&D outcomes (and hence costs) have become common knowledge, rms compete n the product market, ether n quanttes or n prces..1 R&D technology The market s populated by a contnuum of dentcal consumers wth mass equal to 1. A representatve consumer derves the utlty U (q ; q j ), whch s a strctly concave functon, and maxmzes U (q ; q j ) p q p j q j, where q s rm s output and p denotes ts prce. Before rms compete n the product market, they explot cost-reducng R&D opportuntes. 9 Aghon, Gr th & Howtt (006) provde evdence of a U-shaped relatonshp between product market competton and vertcal ntegraton.

6 5 We assume that each rm s ntal margnal cost s c. Ths cost decreases wth a rm s R&D output, y, whch depends on ts agent s e ort, e, and a project-spec c shock, ", takng the form y = e + ". Thus, after the completon of the R&D process, rm ncurs the margnal cost c = c e ". The random term " s drawn from a truncated normal dstrbuton wth zero mean and varance, and les n [, ]. The bound (0; +1) serves to guarantee non-negatve post-nnovaton margnal costs. Thus, gven the R&D technology, agent wll choose an R&D e ort level e Y, where Y [0; c ]. The feasblty lne takes the form c = 0, c e " = 0, mplyng that e c for any decsons of e Y. The shocks, " and " j, are correlated, where j = cov (" ; " j ) s ther covarance and j j denotes the correlaton coe cent, jj 1. The type of correlaton (postve or negatve) may depend on whether the agents use smlar or d erent R&D technologes. For nstance, rms that produce hard dsks may hre researchers that use ether magnetc or holographc technologes. In ths case, a market shock may a ect the output of the projects that are based on these two technologes n a d erent way. Each rm also bears the cost of conductng R&D whch s equvalent to agent s compensaton, w. Thus, rm enjoys the net pro t = Cournot or Bertrand pro t, dependng on the mode of product market competton. w, where s the. Researchers objectves and ncentve contracts To conduct R&D, agent ncurs dsutlty g (e ). Ths functon s twce contnuously d erentable and convex, mplyng that there are dmnshng returns to scale n the R&D producton process. We also assume that g(0) = 0, g 0 (0) = 0 and lm e!1 g 0 (e ) = 1. Agent receves the reward w and has constant absolute rsk-averse (CARA) preferences. She derves utlty U (w ) = e r [w g(e )], (1) where r s the Arrow-Pratt measure of rsk averson. Throughout ths analyss, we assume r j r > 0, mplyng that rm j may hre a hgher rsk-averse agent than rm. 10 Holmström & Mlgrom (1987) establsh that n a model much lke ours, the optmal contracts are lnear. In partcular, agent s compensaton depends lnearly on both agents R&D-outputs due to the correlaton of the market shocks. Relatve performance evaluaton schemes explot all avalable nformaton and allow each prncpal to better perceve hs agent s e ort by comparng both researchers R&D outcomes. Each agent s contract takes the form ( ; ; ) and mples the payment w = + y + y j, () 10 We also show that a postve rsk-ncentves relatonshp can arse even when r = r j = r and the asymmetry n agents payments s orgnated from d erences n the varance of agents R&D output shocks: " has varance, where 6= j. As an agent s ncentves are weakened wth a decrease n jj, the other s ncentves may be strengthened, even f agents are equally rsk-averse.

7 6 where denotes the xed salary component and, are the pay-for-own and pay-for-rval performance parameters, respectvely. If the agent rejects the o er, she pcks the outsde opton, dervng zero utlty. 3 Manageral contracts and Cournot competton We recursvely solve the game where Cournot rvals make ther decsons smultaneously and ndependently. We begn the analyss by consderng a market wth a lnear demand and quadratc cost-of-e ort functons. Then, we allow for general demands and cost functons. 3.1 Equlbrum wth lnear market demand Followng Sngh & Vves (1984), we assume that the representatve consumer s utlty s U(q ; q j ) = h a (q + q j ) 1 q + q j + bq q j, mplyng rm s demand p = a q bq j, where a > c and b [0; 1] s the product d erentaton parameter. The parameter a s hgh enough relatve to the ntal margnal cost, so that each rm nds t best to do some R&D regardless of ts rval s R&D decson. In the downstream market, after the realzaton of the margnal costs, rms compete n quanttes. The level of rm s output that maxmzes = [a q bq j c ] q s and the equlbrum Cournot pro t s = (q ). q = a c + b + y by j 4 b, Each prncpal s the resdual clamant on rm s net pro ts, whch are equal to the expected Cournot pro ts net of hs agent s compensaton. In partcular, gven the belefs about rm j s e ort (denoted by be j ) as a response to rm s R&D level, each prncpal o ers a contract that maxmzes hs net pro ts, whle t guarantees agent s partcpaton and s compatble wth her ncentves to perform. Thus, each prncpal solves the followng constraned maxmzaton problem max f ; ; ;e g E f ( ; ; ; e ; be j ) j " ; " j g = E f w j " ; " j g subject to e = arg max e E fu (w ) j " ; " j g (IC ) E fu (w ) j " ; " j g 0 (IR ) Agent always chooses the R&D e ort level that maxmzes her (expected) utlty, as s captured by the ncentve compatblty constrant (IC ). The ndvdual ratonalty constrant (IR ) serves to guarantee that agent wll stay n the rm and conduct R&D only f by dong so, her expected utlty exceeds her reservaton utlty of zero. Lemma 1 establshes that agents choose the R&D e ort that they would have chosen f the dstrbuton of the shocks was normal but not truncated. Thus, the truncaton of the dstrbuton nsures

8 7 that n the post-nnovaton stage, the margnal costs wll be non-negatve but t does not change agents e ort decsons. Followng Chalot (015), we show that ths property also holds when relatve performance evaluatons are used and the R&D output shocks are correlated. Lemma 1 (Expected utlty wth correlated shocks & truncated normal dstrbuton) Assumng lnear contracts, agent s CARA preferences towards rsk and correlated random terms that follow a truncated normal dstrbuton symmetrc around the mean, then agent s expected utlty becomes where = + r[ + ] ( ) ( E fu (w ) j " ; " j g = j e r[ e U (w )], + r[ + ] ) and j = " p1 +p 1 r " p 1 " p1 +p 1 r " p 1. s the probablty of " fallng nto and the certanty equvalent of her utlty s e U (w ) = E (w ) r V ar (w ) g(e ). The coe cents and j are postve and ndependent of e ort, mplyng that agent chooses the e ort level e that maxmzes e U (w ). Proof. In appendx A.1. To derve explct expressons for the agents ncentves, we also assume that the cost-of-e ort functons are quadratc of the form g(e ) = k e, where hgher k ndcates lower e cency of R&D technology. Thus, agent maxmzes U e r (w ) = + e + e j + j + k e and chooses the optmal e ort e = k. (3) Equaton (3) ndcates that the optmal pay-for-own performance parameter,, wll be postve. agent s hgher R&D output wll be rewarded wth a hgher payment. Followng Itoh (1991), the assumptons of agents CARA preferences and the (truncated) normalty of the random terms as well as the concavty of the functons U and n e allow us to use the rst-order approach. 11 constrant n prncpal s problem. An Thus, equaton (3) replaces the IC The optmal contractual parameters are obtaned by solvng smultaneously both prncpals constraned maxmzaton problems and takng the Kuhn-Tucker condtons. The IR constrant bnds at the optmum, mplyng that the base payment,, s such that guarantees agent s partcpaton, whle prncpal approprates all the surplus. The latter has all the barganng power. Usng equaton (3), we solve e U = 0 wth respect to. Then, we substtute the soluton nto () and take expectatons to derve the expected compensaton 11 Itoh (1991) states that n a mult-agent model, the rst-order approach requres further assumptons n addton to the monotone lkelhood rato property and the convexty of the dstrbuton functon condton (CDFC). In partcular, we need to use a generalzed CDFC for the jont probablty dstrbuton of shocks, the wage schemes must be nondecreasng and the coe cent of absolute rsk-averson must not declne too quckly. In our model wth agents CARA preferences, (truncated) normally dstrbuted random terms, lnear contracts and R&D producton functons, the above requrements are sats ed.

9 8 of agent as a functon of the remanng contract parameters E fw j " ; " j g = 1 + k r k + ( + ) r. (4) For any value of, the optmal s chosen to mnmze the varance of the wage w, =. (5) If the R&D output shocks are postvely correlated, > 0, the optmal s negatve. The prncpal perceves that the researchers perform n a smlar envronment and by settng negatve, he s able to lter out the common shock from hs agent s payment. In fact, the prncpal penalzes hs agent when the rval researcher does better. If < 0, by settng > 0, the prncpal allows hs agent to su er less from a bad outcome and encourages her to perform. The optmal s chosen so that agent s payment s no longer senstve to agent j s R&D output. Usng equatons (3), (4) and (5), the expected net pro t of rm, as a functon of and j becomes 1 E f j " ; " j g = 1 a c b b k b j k (4 b ) 1 + kr 1 ( k ). (6) It conssts of the expected Cournot pro ts ( rst two terms) and the expected compensaton to the agent (thrd term). Each rm chooses the pay-for-own performance parameter that maxmzes ts expected net pro t. By ncreasng, rm balances three forces on ts expected pro ts: ) ts expected margnal cost decreases, ) the expected compensaton to ts manager ncreases and ) ts rval s forced to decrease ts own ncentve parameter, j, and hence produce less output, allowng rm to gan market share. Rvals contractual decsons and j are strategc substtutes. A hgher j (and hence a hgher q j ) nduces rm to lower ts own for two reasons: ) q decreases and so does the bene t from a cost reducton and ) the best-response of rm n the product market when q j ncreases s to become less aggressve by lowerng ts own q (gven that quanttes are also strategc substtutes). Thus, weaker ncentves from rm s the pro t-maxmzng response to an ncreasng j. 13 Solvng for the equlbrum ncentves, we derve = 4 (a c) 4 b ( b) k j 4 k, (7) (4 b ) k h(4 b ) k j 8 8 [(4 b ) k j ] where k 1 + kr 1. We make the assumpton k > 8 (4 b ) to guarantee that the soluton n R&D s nteror and unque. 1 Appendx A. solves the prncpals problems. 13 In subsecton 3., we examne, n a general-demand case, the underlyng e ects that determne the nature of rvals strategc nteractons n R&D (strategc substtutes or complements). In addton to the two e ects that we just dscussed n the lnear demand case, there s a thrd e ect that arses due to the curvature of the demand.

10 Relatonshp between rsk and R&D ncentves When the agents have the same degree of rsk averson, r = r j reduces to = j = 4 (a c) k (4 b ) ( + b) [1 + kr (1 )] 4. = r, the equlbrum pece-rate pay Clearly, hgher rsk, as t can be measured by an ncrease n (or a decrease n the degree of correlaton jj), wll nduce both prncpals to weaken the ncentves they provde to ther agents. We argue that ths standard result n the lterature may not hold when agents are heterogeneous wth respect to ther rsk tolerance and rms compete n the product market. Proposton 1 establshes that due to product market competton, the negatve relatonshp between ncentves and rsk can be reversed for the rm wth a less rsk-averse agent. 14 To reduce the length of the expresson n the next Proposton and wthout loss of generalty we set b = 1. Proposton 1 (Relatonshp between rsk and ncentves) Suppose ncreases and r < r j. The rm wth the hgher rsk-averse agent weakens the ncentves to hs agent, d j d < 0. In contrast, the rm wth the less rsk-averse agent strengthens the ncentves to hs agent when the bene ts n pro ts from busness stealng exceed the cost of provdng nsurance: d d r < Proof. In appendx A.3. > 0, f and only f, 4r j (3k 4) 3 + 3k (9k 0) + 9k 3 r j (1 ) [3k 3 r j (1 ) + (3k 4)]. Agent j abhors rsk to a greater extent compared to agent, seekng more nsurance. As ncreases, the moral hazard problem wll become even more severe for the hghly rsk-averse agent j, snce her payment s now noser and more senstve to market shocks. She requres addtonal nsurance that can be provded wth a further decrease n R&D ncentves (nsurance e ect). As rm j nvests less n R&D, t does not gan as much n cost reducton and decreases ts producton. Thus, there s room for busness expanson (busness stealng e ect). Although rm also has to face the moral hazard problem, ts agent responds less to an ncrease n rsk. If the cost of exertng e ort s small enough, rm may nd t optmal to provde even stronger ncentves to her agent and exert hgher e ort n order to explot the avalable opportuntes n the product market and expand ts busness. The ntuton behnd Proposton 1 s best captured by Fgure 1. Incentves between the two rms are strategc substtutes. Hgher rsk shfts both reacton curves nwards, mplyng that when we hold the ncentves of the rval xed, the rm n queston optmally weakens ncentves. But rsk shfts the reactons curves n d erent magntudes across the two rms. The rm wth the more rsk-averse agent, rm j, experences 14 Equatons (3) and (5) show that there s one-to-one relatonshp between and optmal e ort, e, as well as the relatve performance parameter,, that s used to lter out the common uncertanty about researchers R&D performances. Thus, ndrectly, our dscusson sheds also nsghts on the changes of the optmal e orts.

11 10 the bgger shft, thereby lowerng ts ncentves sgn cantly. Ths s a commtment to decrease output n the next stage, whch presents an opportunty for rm, whose cost of ncentvzng ts agent s relatvely low, to strengthen ts ncentves n order to gan market share. 15 Fgure 1: Change n ncentves as common rsk ncreases. Intal equlbrum s at A and nal equlbrum at B. Frm strengthens ts ncentves, whle rm j weakens them. To shed addtonal nsghts nto Proposton 1, we conduct a Slutsky-lke analyss to dscuss the underlyng e ects of on rvals optmal ncentves. We rewrte the rst-order condtons wth respect to and j by usng equatons (3) and (5), and then, we totally d erentate both rvals rst-order condtons wth respect to. In partcular, provded that " and " j le n, we take d(@ef g=@ g=@ Ef g d = 0. Ths yelds d = 0, that mples j 4b d (4 b ) k d {z } busness stealng e ect 4 d (4 b ) k d {z } busness stealng e ect r 1 {z } nsurance e ect r j 1 j {z } nsurance e ect + {z} <0 by the SOC + j {z} <0 by the SOC d = 0, (8) d d j = 0. (9) d Frm s pro t functon s concave n and the second-order condton (SOC) of the constraned maxmzaton problem holds, mplyng Ef g 8 = (4 b ) s always negatve. Equaton (8) shows the e ects of on rm s optmal ncentves, whle equaton (9) decomposes the e ect on rm j. Let us rst suppose that the busness stealng e ect s absent because each rm s a monopoly. A hgher a ects both and j negatvely. Then, allow for strategc nteractons wth respect to ncentves. Upon nspecton of (8) and (9), we nfer that only one of the followng two possbltes can arse: ) ether rsk 15 We o er a numercal example that sats es the condton of Proposton 1. Let a = 100, b = 1, c = 45, k = 1:, = 1 and = 0:1. If r = 0:1 and r j = 0:4, then rm provdes stronger ncentves to ts agent as rsk ncreases.

12 11 a ects ncentves negatvely n both rms, ) or postvely n one rm and negatvely n the other. When the asymmetry across rms wth respect to the degrees of rsk averson of ther agents s sgn cant -.e., r j s hgh enough relatve to r - the nsurance e ect s small n rm and large n rm j. Let us assume that rm s nsurance e ect s arbtrarly close to zero. Then, from (8), d d and d j d nsurance e ect n rm j s strong enough, then, from (9), we have d j d cannot have the same sgn. If the < 0, whch mples that d d > 0. Ths result holds even f agents are equally rsk averse, r = r j = r, and the asymmetry n agents compensaton s orgnated from the d erent varances of agents R&D output shocks. Suppose that R&D output shocks are dosyncratc and asymmetrc, < j. Agent j s R&D output s noser. The level of rsk to whch agents are exposed now ncreases as jj gets smaller -.e., the nsurance e ect s a decreasng functon of jj. As jj decreases, relatve performance evaluaton schemes as means of lterng out the common shock from agents payment, become less e ectve. 16 The rm whose agent s R&D output s noser provdes weaker ncentves and becomes softer n the product market. In turn, the rm that s exposed to lower dosyncratc rsk can strengthen the ncentves to conduct R&D and hence extend ts busness vs-à-vs ts rval. Ths model establshes that manageral ncentves respond to a change n the corporate envronment common to all rms n the ndustry - e.g., systemc rsk - n a fashon that dspels tradtonal agency theory. Ths can also happen as manageral ncentves respond to a change n a rm s spec c corporate envronment - e.g., dosyncratc rsk. When only the rsk to whch one rm s exposed changes (say ), then the e ect on ncentves must be asymmetrc across the two rms: the equlbrum ncentves and move to opposte drectons. Ths s because only rm s best-response curve shfts nwards (see Fgure ). 17 As ncreases, rm j wll always provde hgher-power ncentves to hs agent, for any r, r j, and j. Thus, f market changes a ect only one rm, the standard result n the lterature never holds. 16 Equatons (8) and (9) become, respectvely, 4b dj (4 b ) k 4b dj (4 b ) k d + djj r + d = 0 and djj 4b d (4 b ) k + djj r j j + j d j = 0. djj 17 Equatons (8) and (9) become, respectvely, r 1 d j + 4b d = 0 and + j d d = 0. d There s no e ect of on the provded nsurance to agent j, whle the busness stealng e ect arses due to rms nteractons n the product market. We have d d < 0 whle d d j > 0. (4 b ) k d

13 1 Fgure : Change n ncentves as dosyncratc rsk n rm ncreases. Intal equlbrum s at A and nal equlbrum at B. Frm weakens ts ncentves, whle rm j strengthens them. Ths analyss bols down to the followng: n ndustres where competton s ntense and thus busness stealng ncentves are strong such as n mcroelectroncs-based ndustres, pharmaceutcals, or even n the nancal sector as rms are exposed to hgher rsk, they may not adjust ther pay-for-performance R&D ncentves n the same drecton. Asymmetres on the part of agents preferences towards rsk or on the varance of R&D producton shocks play a key role. What drves the result s the strategc bene t of the rms compared to the cost of provdng nsurance to ther agents. 3. R&D ncentves wth general market demand We now show that for any functon of market demand and dsutlty of labor, rsk can ncrease ncentves as long as the R&D decsons are strategc substtutes. We adopt the prevous setup where rms appont agents wth d erent degrees of rsk averson, r < r j, and the level of rsk changes wth. Increases n wll n uence both rms R&D best responses. Instead, one can perform a smlar analyss by consderng that rms whose agents have the same preferences towards rsk to provde a d erent level of nsurance only because and j are d erent. Ths analyss follows the same logc, and thus t s omtted. To conduct R&D, we assume as n subsecton (:) that agent ncurs dsutlty g(e ). A general utlty functon U (q ; q j ) mples the nverse demand system p = d (q ; q j ) for any and j. Ths functon s < 0, and the cross dervatves are < 0, gven that the quanttes of Cournot rvals are strategc substtutes. An ncrease n a rm s output s also somewhat more e ectve n determnng ts own market prce than ts @q > Frm s pro t s c = c w, where the Cournot pro t now s c (d c ) q. The superscrpt c ndcates the values n a settng wth general demand functons and Cournot competton n the downstream market. The followng assumptons on the pro t functons also hold. < 0 for any, j. < 0 for any, j. < 0 for any, j. Assumpton (C:1) ensures that each rm s pro t functon s strctly concave n ts own output. Assumpton (C:) guarantees that rms reacton functons n the product market are well-behaved and ther slopes are less than one. In turn, t ensures that n the producton stage, there exsts a unque nteror Nash equlbrum n quanttes. Assumpton (C:3) guarantees that rvals quantty decsons are strategc substtutes. De nton 1 (Degree of substtutablty of products under Cournot competton) Under assumptons (C:1) (C:3), rms products exhbt decreasng (ncreasng) substtutablty, f an ncrease n a rval s producton dmnshes a rm s pro t at a decreasng (ncreasng) j > (<) 0 for any, j.

14 13 Accordng to De nton 1, for decreasng substtutablty, the demand for rm s product needs to be a convex functon of q j j > 0. As q j ncreases, the two products become weaker substtutes,.e., the negatve e ect of q j on d (the prce of good ) becomes smaller. Smlarly, for ncreasng substtutablty, the demand functon of rm n ts rval s output s requred to be d < We recursvely solve for the equlbrum. In the last stage, rms observe the realzaton of the margnal costs and compete for consumers. Each rm maxmzes ts pro t functon c = [d (q ; q j ) ts rst @q q +d c ] q, by takng c = 0. Rvals solve these condtons and choose the equlbrum levels q c and qc j. Then, we proceed n dervng the contractual choces that nduce the optmal e ort level. Equaton (5) whch shows the relatonshp between and stll holds and the optmal e ort sats es g 0 (e ) =. Lemma hghlghts the e ects of the R&D ncentves,, on both rms equlbrum outputs. Lemma (E ects of R&D on optmal outputs under Cournot Competton) Under assumptons (C:1) (C:3), a rm s equlbrum output s ncreasng n ts own agent s R&D ncentves and decreasng n ts rval ncentves: > < 0 for any and j. Moreover, q c j and qj separable n and j, = 0. j are lnear functons and addtvely Proof. In appendx A.4. In the nnovaton stage, the level of R&D conducted by the rms depends on the strategc propertes of agents R&D ncentves. Thus, we need to specfy the condtons under whch ncentves are strategc substtutes or complements. We begn by examnng rm s rst order condton wth respect to. We consder that the followng dervatves are n expected terms, snce the decsons are taken before the realzaton of the R&D output shocks. Usng the envelope theorem, a ects c through q j, the margnal cost c and the agent s expected @q c @ = 0. (10) The rst term s the strategc (ndrect) e ect of whle the second and thrd terms capture the drect e ect on rm s pro ts that arses even n the absence of product market competton (Fudenberg & Trole (1984)). The drect e ect s postve because the acton (hgher and hence hgher q ) s tough and the goods are strategc substtutes. Agent s R&D ncentves respond to a change n j as follows We also assume that d d < 1 n order to obtan a unque equlbrum j c ; c j.

15 14 d d j = H c, where j {z } > c c {z } < 0 {z } < {z } > c @q {z {z } < 0 > 0 Decr. (ncr.) Lemma Lemma substtutablty and @ @ < 0 follows from the second-order condtons. There are three (possbly) opposng e ects that are present and determne the nature of rvals strategc nteractons n R&D. The rst e ect n (11) s absent when demand s lnear and addtvely separable. The other two e ects are those we dscussed n the lnear demand case whch are sources of strategc substtutablty. 19;0 However, the presence of the rst e ect can change the nature of rvals strategc nteractons. It depends on the degree of substtutablty between rms goods and how t changes when q j changes. Due to product market competton, busness expanson from rm requres the market share of rm j to shrnk. If demand exhbts decreasng substtutablty, such strategc nteractons wll bene t rm sgn cantly. Wth more d erentated products, rm has more to gan by strengthenng the R&D ncentves that provdes to ts agent. In turn, rm j serves a smaller market share and thus needs to exert lower R&D e ort. Under ncreasng substtutablty, the rval s response to rm s busness stealng does not bene t the latter as much. The busness stealng ncentves are present, but each unt of R&D conducted by a rm s less e ectve n securng ts strategc poston n the downstream market. Thus, rm does have ncentves to nnovate, but such ncentves become progressvely weaker as the R&D-takng rm has less to bene t from cost reducton. Ths s a source of strategc complementarty. Lemma 3 spec es the condtons under whch ncentves are strategc substtutes. 1 Lemma 3 (Strategc Interactons n R&D under Cournot Competton) Under assumptons (C:1) (C:3), researchers R&D ncentves, and j, are strategc substtutes f demand functons exhbt decreasng or weakly ncreasng substtutablty: d d j Proof. In appendx A.5. < 0 f and only @ c @q 19 The second e ect depends on the nature of rvals strategc nteractons n the producton stage. Gven that quanttes strategc substtutes, < 0, rm ncentvzes ts agent to do more R&D n order to ncrease ts producton. Ths move wll nduce a medocre" output response of rm j. The latter rm now serves a smaller market share and thus, t decreases j. 0 The thrd term captures the e ect of j on the bene ts a rm approprates from e cency enhancement. For lower j, rm j becomes softer n the product market, strengthenng rm s strategc c < 0. Due to the strategc between rvals outputs, rm bene ts more from nnovatng tself n order to nduce a cost < 0. Frm now produces more to serve more consumers and thus a further decrease n rm s margnal cost contrbutes more to rm s pro tablty. Thus, rm responds to a decrease n j by ncreasng. 1 The result n Lemma 3 s remnscent of Vves (009), where, n Secton 3.1. of hs paper, he presents a determnstc model of capacty nvestments n a Cournot duopoly. One of the su cent condtons for cost-reducng nvestments to be strategc substtutes s what we call n our paper decreasng substtutablty. Another su cent condton s q ( ; j ) to be submodular n ( ; j ), whch s also true (actually, t s both submodular and supermodular) n our model (see Lemma ).

16 15 Holdng j constant, an ncrease n shfts rm s R&D best-response curve nwards. The trade-o between ncentves and nsurance s shfted towards the latter and gven j, rm provdes weaker ncentves to hs agent. Due to rsk-sharng, each prncpal sets a lower pay-for-own ncentve parameter and exerts less e ort as the systematc rsk ncreases, all else beng equal: d c d = 1 c r 1 g 0 (e ) g 00 (e ) < 0, (1) where e c s determned by c. However, n ths strategc envronment, all else cannot be equal, snce each rm wll optmally respond to any change n her rval s decson. We show that, n equlbrum, as ncreases, a rm may even provde stronger ncentves and exert hgher e ort, provded that rms R&D decsons are strategc substtutes. For ths to happen, agents also need to be substantally heterogeneous n terms of rsk tolerance. The decomposton of the e ects of on both rms optmal R&D ncentves gves d c j H j d r 1 c g 00 (e c ( c )) + c = 0, (13) d d c H d r j 1 c jg 00 e c j c d c j + c j j = 0. (14) d Agent s e ort e c s a (postve and monotonc) functon of c, and the coe cents H, c d c are gven by (11). Equatons (13) and (14) show the e ects of on rm s and rm j s optmal ncentves, respectvely. They are also the analogues of (8) and (9) from the lnear demand and quadratc cost-of-e ort case. Gven that dc j d < 0, equatons (13) and (14) show that, for hgher, rm wth the less rsk averse agent acqures more R&D n equlbrum, dc d > 0, as long as rvals R&D decsons are strategc substtutes - H < 0 and H j < 0 - and the bene ts from busness stealng are substantal to exceed the agency cost. In partcular, when H j < 0, the busness stealng e ect n equaton (13), s postve. Proposton establshes that rsk and ncentves can be postvely related for the rm that s less exposed to rsk for any demand and cost of e ort functons, as long as rvals R&D decsons are strategc substtutes and busness stealng ncentves are strong. Proposton (Strategc nteractons n R&D wth general demand functons) Under assumptons (C:1) (C:3), suppose that r < r j. If rvals R&D decsons are strategc substtutes, a hgher weakens the optmal R&D ncentves of the rm wth the hgher rsk-averse agent, whle the rm wth the less rsk-averse agent wll provde stronger ncentves as long as the bene ts n pro ts from busness stealng exceed the cost of dong R&D: dc d > 0 and dc j < 0 f and only f c r (1 d c j r j(1 )g 00 (e c (c)) )g 00 (e c j( c j)) < H j c, where H j and c j are gven by j equaton (11). Ths expresson does not descrbe the change n equlbrum ncentves; rather, t descrbes how a rm s R&D best-response curve shfts.

17 16 Proof. In appendx A.6. If R&D decsons are strategc complements, H > 0 and H j > 0, rsk always decreases e ort for both rvals, as s standard n the lterature. Gven that dc j d < 0, the busness stealng e ect n equaton (13) s always negatve. Both e ects move to the same drectons leadng rm also to underprovde R&D ncentves as ncreases, dc d < 0. Equaton (14) s also sats ed only when hgher decreases both c and c j. The rsk-e ort relatonshp becomes postve only when both rms R&D reacton curves are downward slopng and the cost of dong R&D s small enough. 4 Moral hazard and Bertrand Competton We now analyze the rvals strategc decsons and the role of nsurance provson when rms compete à la Bertrand n the product market. We show that a postve rsk-ncentves relatonshp can be obtaned regardless of the mode of competton (Cournot or Bertrand) n the product market, but n the R&D market, rvals decsons need to be strategc substtutes. We rst dscuss the lnear demand case, and then develop our arguments by consderng general demand and cost functons. 4.1 Lnear market demand and optmal R&D decsons The product market nvolves competton n prces. Rewrtng the Cournot demand wth respect to prces, we get q = p + p j, where a(1 b) 1 b, 1 1 b and b 1 b. We rst solve the Bertrand game and obtan rm s equlbrum prce: p = 1 4 [ ( c ) + ( + c j )]. Thus, rm s R&D allows t to set a lower prce. After the realzaton of margnal costs, rm s Bertrand pro t s q = 4 ( + ) = 1 (q ), where c + c j. Then, we derve the agents optmal R&D ncentves. In the R&D and contract stage wth cost-of-e ort functon k e, the equatons (3), (4) and (5) hold, and lead to the equlbrum pay-for-own performance parameter of the form = [ c ( )] 4 ( + ) k j ( + ) 4 h 4 k k j v, where v 4 k. D erentatng both agents equlbrum ncentves wth respect to, one can ascertan that hgher rsk can ntensfy the R&D ncentves of the rm wth the less rsk averse agent, even f the product market rvals are engaged n Bertrand competton. For example, lettng = 0:95, k = 1:5, = 0:1, r = 0:4, r j = 0:1, = 1, a = 100 and c = 45, we get opposng e ects of an ncrease n on Bertrand rvals R&D ncentves, d d > 0 and d j d < 0. Ths result always holds n settngs wth lnear demands, because rvals R&D bestresponse curves are downward slopng. Frms decsons n the R&D market do not nhert the propertes of the strategc varables (prces) beng utlzed n the product market.

18 17 As n the lnear demand Cournot model, there are two forces that determne the nature of rvals strategc nteractons n R&D, but these forces are now opposng. On the one hand, we consder changes as j ncreases. If rm cannot adjust ts prce, the ncrease n rm j s R&D wll lead to a drop n her own prce, takng away busness from rm. The bene ts from margnal cost reducton dmnsh for rm, nducng her to provde weaker ncentves and exert lower R&D e ort. Ths e ect s a source of strategc substtutablty between rvals R&D decsons. One the other hand, when rm lowers ts prce n response to the prce reducton by rm j because prces are strategc complements, rm s sales go up, creatng an ncentves for rm to nnovaton more. Ths can be acheved by nducng her agent to devote more e ort nto the R&D process, whch requres stronger ncentves. Ths e ect captures changes as j ncreases and s a source of strategc complementarty n rvals R&D. Wth lnear demand, the former e ect that dscourages rm to acqure R&D as a response to hgher j always domnates: the slope of rm s R&D-best response lne d depends on the j j @ = ( ), whch s negatve. Thus, R&D (4 ) best response lnes are downward slopng and rvals R&D decsons are strategc substtutes: a hgher j wll be assocated wth a lower. Therefore, we argue that even f rvals compete à la Bertrand n the product market, gven that ther R&D decsons are strategc substtutes, rsk can ncrease the ncentves n the rm that needs to provde less nsurance to ts agent, as n the Cournot model. In partcular, we argue that the nsurance-ncentves trade-o s more crtcal for agent j whose payment s more senstve to rsk. Thus, her R&D ncentves are dstorted downwards, mplyng a medocre cost reducton and a loss n market share. The rval rm that needs to provde less nsurance can take advantage of busness stealng when the gans n market share exceed the cost of moral hazard. As rsk ncreases, the bene ts for the latter rm from acqurng more R&D and expandng ts busness are augmented. 4. General market demand and Bertrand rvals ncentves The representatve consumer s maxmzaton problem gves rse to a general demand system q = D (p ; p j ) for any and j. The drect demand functons are < 0, and the cross-dervatves j > 0. The own-prce s also larger than the cross-prce j. Frm s pro t now s gven by b = b w, where b (p c ) D s the Bertrand pro t. The superscrpt denotes the choces of Bertrand rvals when they face general demand functons. The followng assumptons on the pro t functons are also n order. b < 0 for any, @p j < 0 for any, j > 0 for any, j. Smlarly to the Cournot case, assumpton (B:1) ensures that a rm s pro t functon s strctly concave

19 18 n ts own prce. Assumptons (B:) and (B:3) guarantee the nterorty and unqueness of the equlbrum n prces. De nton (Degree of substtutablty of products under Bertrand competton) Under assumptons (B:1) (B:3), rms products exhbt ncreasng (decreasng) substtutablty, f an ncrease n the rval s prce rases a rm s pro t at an ncreasng (decreasng) j > (<) 0 for any, j. For ncreasng substtutablty, the demand for rm s product needs to be convex n ts rval s prce, > 0, whle a concave demand functon n p j s requred for decreasng substtutablty between rms j functons. < 0. In the Bertrand case, we also need to consder the degree of modularty of the demand De nton 3 (Degree of modularty of demand functons) Under assumptons (B:1) (B:3), f an ncrease n a rval s prce strengthens (weakens) the e ect of a rm s prce on ts own market demand, ths demand functon exhbts supermodularty (submodularty) wth respect j > (<) 0 for any, j. s=(s 1) s=(s 1) s=(s 1) For example, consder the demand functon q = Ip = p + pj whch s derved from a constant elastcty of substtuton (CES) utlty. We have s D 1; 1] and I denotes the consumer s ncome. Ths demand functon sats es ncreasng substtutablty and supermodularty when s = decreasng substtutablty and submodularty when s = 1. 1, whle t exhbts To shed addtonal nsghts, we need to examne the strength of strategc complementarty between rms b j = (p c D j. For nstance, f a rm s demand s supermodular wth respect to prces, an ncrease n a rval s prce bene ts the rm: the hgher s the prce-cost margn (markup), the more a rm bene ts as ts rval s product becomes more expensve. Thus, by nvestng n cost-reducng R&D, a rm becomes more e cent and better able to take advantage of a rase n a rval s prce, snce the rval wll lose market share. If demand s lnear and addtvely separable, we D j = 0. We solve the game backwards and derve the equlbrum prces p b ; j and p b j ; j. In the R&D and contract stages, rm chooses the R&D ncentves that maxmze ts expected Bertrand pro t net ts agent s wage. Lemma 4 establshes the e ect of an agent s ncentves through on ts own rm s and ts rval s equlbrum prces as well as the cross e ects. In fact, as a prncpal o ers a contract that nduces hs agent to nvest more n R&D, he realzes a lower margnal cost, allowng hm to set a lower prce. Gven the strategc complementarty of prces, the rval rm s eager to lower ts own prce as well. The followng analyss presupposes h that the values are n expected terms. Note also = @p = 1 (p j c j D @ b @p b > 0 s mpled from the stablty condton. Gven that n the latter derv- atve only n uences the margnal cost, we p b = @p. (15)