University of School of Economics

Conference Version for AsLEA 2014 Annual Conference

An Economic Model for Structures of the Legal Profession

Ashley Cheng Hodaka Morita School of Economics School of Economics UNSW Australia UNSW Australia An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Abstract

This paper examines a puzzling observation about the state of the legal profession. Why do some jurisdictions have a fused profession system whilst other have a split profession system? This paper explores some of the trade-offs involved in adopting one system over the other. We develop a theoretical model for litigation and use it to analyse outcomes of cases under the fused profession system and the split profession. In particular, we draw conclusions about the efficiency of the allocation of effort, the strength of incentives and its effect on lawyers and how the structure of the legal profession affects the probability that the legal system produces the right outcome. We show that, where the party who ought to win has the higher stakes in the litigation, a fused profession will produce the correct result more often. However, the split profession is better when the stakes are more equal. We also draw some important implications about -making and the structure of the legal profession. In particular, we illustrate a link between the structure of the legal profession and the correlation between which party ought to win and which party has the larger payoff from winning. This link provides one way to examine law making and has some important implications to legal policy.

Note: This paper is a working paper. In the interests of clarity and brevity, the mathematical proofs have been omitted in this conference version. However, for readers’ interest, a section of Appendix A has been included to illustrate the proof for Proposition 1. The corresponding author (and presenter) is Ashley Cheng ([email protected])

1 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

1 Introduction and Motivation

∼ The division of the profession into two distinct branches is one of the most obvious peculiarities of our system, and requires to be rationally and convincingly justified ∼

- Evershed Report, 1953 1

1.1 Introduction

In 1937, Ronald Coase asked the following question: If markets are so efficient, then why do firms exist? (Coase, 1937). Coase answered that markets were not always efficient and that there are trade-offs between markets and firms. This paper asks a question in a similar vein: Why do different jurisdictions adopt different methods of organising their legal professions? The simple answer to this question is that there must be trade-offs involved in structuring the legal profession as a Split Profession compared with a Fused Profession. Thus, under different circumstances, one structure might perform better than the other. The aim of this paper is to provide some explanation of the trade-offs and draw some implications about how the structure of the legal profession affects outcomes in the legal system.

There are two predominant ways to organise the legal profession: 1) A Split Profession which distinguishes between and ; and 2) a Fused Profession which does not. That different jurisdictions use different systems, and persist in using those systems, suggests that there are trade-offs between the two systems. However, those trade-offs have not been well studied. This paper presents one possible methodology for understanding those trade-offs. We employ a theoretical model of litigation to analyse the impact of the two different systems on litigation outcomes. The client hires the lawyers as agents and incentivises them according to binary incentive contracts. To the best of our knowledge, this approach to modelling litigation is unknown to the literature. We assess the merits of each system on two criteria: 1) Efficiency and 2) the probability with which justice is realised. We then derive implications for policy making arising from pre- existing structures of the legal profession.

For clarity, we explain the specific terminology used in this paper. ‘Lawyer’ refer to any legal practitioner without distinguishing between attorneys, solicitors and barristers. The word ‘attorney’ is used in the sense of an attorney-at-law specifically referring to a lawyer practising in a Fused profession. ‘’ and ‘’ refer specifically to those two types of lawyer. We use the terms ‘Fused Profession and ‘Attorney system interchangeably. Similarly, the terms ‘Split Profession and ‘Solicitor-Barrister system are interchangeable.

A Split Profession system is characterised by the division of labour and responsibility for the conduct of litigation amongst two classes of lawyers: solicitors and barristers. A solicitor is “a class of legal practitioners, generally responsible for advising clients on legal matters, preparing legal documents, representing clients in summary matters and instructing barristers in relation to more complex advocacy work” (Butt, 2006). Solicitors are responsible for the preparatory work necessary to conduct litigation. On the other hand, barristers are professional advocates who are, “by law or custom limited to advocacy and advisory work” (Butt, 2006). When a client is faced with a legal issue and requires representation in litigation, he/she will approach a solicitor. The solicitor will make initial observations on the case and, should the case raise complex issues of law, engage

1Cmnd 8878, Evershed and Ors. (1953)

2 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita a barrister to assist. Whilst the Client is ultimately responsible for paying the barrister’s fees, the contract for services exists between the solicitor and barrister. As a result, even if the Client does not pay the solicitor, the solicitor is still responsible for paying the barrister. Furthermore, because the barrister contracts directly with the solicitor, the barrister does not usually deal directly with the client. In the rare circumstances when barrister and client meet, the solicitor is invariably present Ross and MacFarlane (2007). As a result, the direct contact between the barrister and his client is minimised. This arrangement preserves the independence of the barrister from his/her client.

The barrister is responsible for the conduct of the case in Court. The barrister will prepare written submissions to the Court and augment those written submissions with oral argument. In cases which go to trial, the barrister will also present the evidence as well as examine and cross- examine witnesses. Under most Split Profession systems, barristers are subject to strict rules of conduct. One important rule, for example, which binds the conduct of barristers is the barrister’s paramount duty to the Court. A manifestation of this principle is Rule 25 of the NSW Barrister’s Rules and Rule 708 of the UK Code of Conduct. These rules impose the duty of frankness and candour upon the barrister by requiring that the barrister disclose any relevant precedent (i.e. a previously decided case) to the Court even if that precedent goes against the barrister’s own case. This rule emphasises the arm’s length nature of the barrister-solicitor-client relationship in that the barrister must be independent from the client.

A number of countries such as Australia (but only in New South Wales and ), England and Wales, South Africa and , employ the Split Profession system. Similar systems are also employed in Spain, where Procuradores mirror the role of the solicitor whilst Abogado mirror the role of the barrister, and in civil law countries such as Japan and France. Furthermore, in some jurisdictions which utilise a Fused Profession system, there may be an informal split whereby some practitioners specialise in roles akin to barrister and others specialise in roles akin to solicitors. This de facto arrangement is observed in countries such as Australia ( and ), Canada (excepting Quebec) and to some extent in the United States.

A Fused Profession structure is one where there is no distinction between solicitors and barristers. Instead, legal practitioners are known as either attorneys-at-law, advocates or solicitor-advocates. By its strictest legal definition, an attorney is a “person appointed by another to represent or act in place of that person” (Butt, 2006). In the context of the Fused Profession system, the term ‘attorney’ is in fact a contraction of ‘attorney-at-law’ meaning a person appointed by another to represent/act in the place of that other with respect to his/her legal affairs. That is, an attorney acts the client’s legal representative. The attorney is responsible for conducting the necessary preparatory work and for advocating the client’s case in Court. When a client is faced with a legal issue and requires representation in litigation, he/she will approach the attorney directly. The attorney will take conduct of the matter from start to finish. The client pays the attorney’s fees directly to him/her.

Proponents of the Split Profession system argue that it promotes the independence of the barrister (who is freer of conflicts of interest compared to the solicitor), takes advantage of the division of labour and the advantages of specialisation, and promotes the interests of justice (Commission, 1982). A Split Profession system means that the barrister has less contact with the client.

3 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Indeed, the barrister does not develop any long-term relationship with the client 2 which prevents the barrister from being swayed by concerns over future relationships with the same client. Furthermore, the nature and volume of the work carried out by legal practitioners can necessitate the division of responsibility for different tasks amongst lawyers. The Split Professions facilitates this division of labour and can improve the efficiency of the system because each type of lawyer can specialise in his/her role (Commission, 1982).

The disadvantages of the Split Profession system are often the same advantages offered by proponents of the Fused Profession system. Critics of the Split Profession system argue that it increases the cost of obtaining legal services and, consequently, limits access to legal services (Commission, 1982). Furthermore they argue that there is too much rigidity in the system which can lead to inefficiencies (Commission, 1982).

The arguments in favour of a Fused Profession system are complementary to the criticism of the Split Profession system. Advocates of the Fused Profession system question the need for a distinction between solicitors and barrister (see, for example, (Cohen, 1987)). They consider that “in many cases, two lawyers are used where one lawyer would be sufficient and more economical” (Commission, 1982). Furthermore, they argue that the division of labour in the Split Profession system is not efficient which results in duplication of work (Commission, 1982). As such, proponents of the Fused Profession believe that the fusion of solicitor and barrister will improve social outcomes by eliminating the inefficiencies caused by the distinction.

2 Literature Review

There are very few papers which analyse the structure of the legal profession from the perspective of the trade-offs between a Fused vs. Split profession. The legal jurisprudence analysing the two systems for organising the legal profession is dominated by the lens of law reform. Most papers, such as Maute (2002) accept the Split Profession system as the default position and discuss the flaws in that system. Maute finds that the Split Profession system in England was a ‘result of historical accident, driven by class distinction and economic turf protection’ (Maute, 2002, p. 1358) concluding that there is no reason that the English legal profession could not be fused. However, she does note that complete fusion would be unlikely because there will be a need for specialist advocates.

Cohen (1987) takes a similar position to Maute but goes further, making the argument that there are significant costs associated with the Split Profession which are not justified by the benefits. In particular, he argues that there is a waste of talent because of frictions in moving between the two professions. Furthermore, Cohen argues, the restrictive admission criteria to become a barrister may deter suitable applicants resulting in a waste of human talent. In support of his argument, Cohen follows Reeves (1986) who strongly criticises the Split Profession system. Reeves argues that the legal profession is too rigidly divided and cannot adapt to changing consumer needs and demands. He states that there is an overall cost savings to fusion which result from avoiding wastage of talent and duplication of work.

Forbes (1979), on the other hand, argues favourably for a Split Profession system. Forbes presents

2see Remarks during Swearing-in Ceremony [1952] 85 CLR XI-XII per Dixon CJ

4 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

five arguments in support of the Split Profession. Firstly, he makes the well accepted argument that the Split Profession takes advantage of the returns to specialisation. Secondly, he argues that there are no significant cost savings achieved by the Fused Profession system. Forbes goes further, and claims that ‘if division is costlier, the extra cost may be worth paying’ (Forbes, 1979, p. 253). Forbes makes the argument that ethical standards are raised through the use of the Split Profession because the barrister is separated from the client by the solicitor. He argues this preserves the independence and detachment of barristers. Finally, Forbes argues that the Split Profession system leads to better legal education and scholarship (Forbes, 1979, p. 264).

On the other hand, New South Wales Law Reform Commission (1982) Report, on the legal profession, makes balanced arguments for and against the Split Profession system. The Commission notes the common arguments made in favour of the Split Profession: 1) there are gains to specialisation; and 2) the lawyers are more independence and objective in the execution of their duties. Furthermore, the Commission notes that there are circumstances where the Split Profession achieves better outcomes because there is a division of labour (Commission, 1982, para. 3.45- 3.50). The Commission also notes that the Split Profession reduces ‘client-poaching’ and promotes collegiality at the Bar (Commission, 1982, para. 3.69, 3.74-3.75). However, the Commission ultimately recommends a Fused Profession system because it found that, in the majority of cases, ‘one lawyer would be sufficient and more economical’(Commission, 1982, para. 3.80) Despite the legal jurisprudence, there is a paucity of economic scholarship which examines the differences and trade-offs between a Split Profession and a Fused Profession. Bishop (1989) conducts an analysis of the Split Profession and the Fused Profession system through a vertical integration model. Bishop states that ‘the evidence of most jurisdictions where it is allowed that vertical integration tends to emerge naturally in the market for legal services. So, prohibiting it does generate some cost...Is that cost offset by some larger gain?’ (Bishop, 1989, p. 329). Bishop argues that a Split Profession reduces the agency costs involved in the client-solicitor relationship and takes advantage of returns to specialisation as lawyers are forced to specialise as either solicitors or barristers Bishop claims that there are two beneficial externalities created by the Split Profession. Firstly, there is ‘public capital-formation’ because better quality precedent cases are litigated (Bishop, 1989, p. 332). If barristers acquire highly specialised knowledge, this enhances the quality of their legal argument. Furthermore, since judges are primarily drawn from the ranks of barrister, the quality of judicial officers is enhanced by the specialist knowledge they acquire as a barrister. The second positive externality of the Split Profession is lower costs of administering justice. The lower costs arise because the specialist barrister is able to put a better case to the judge. As such, the judge is better informed and spends less effort making the correct decision.

On the other hand, Bishop argues that there is a potential negative externality in the Split Profession because it encourages the formation of clubs. If ‘the standards of the “club” are stronger than the commercial self-interest of the lawyer as businessman’ (Bishop, 1989, p. 331), then the standards of the club can positively influence the behaviour of the barristers. This may lead to higher ethical standards amongst members of the club. On the other hand, if the standards of the club are weaker than the self-interest of the lawyer, there may be decreased ethical standards because the club has a negative influence. Bishop states that the trade-offs between the two systems makes the choice unclear.

5 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

3 A Formal Model 3.1 The Model

In this section, we present a model which replicates litigation between two parties and use it to compare outcomes from the attorney system and outcomes from the solicitor-barrister system. Suppose there are two parties (call them Clients) who are opposing each other in litigation. Let

πi ≥ 0 be the payoff to Player i for i = {1, 2}from winning the litigation and normalise the value of losing to zero. Let each Client have, what we will call, a Justice Value which we denote with zi where zi ≥ 0. A Client’s Justice Value represents the inherent justice of that client’s case. In one sense, the Justice Value represents how much society thinks that particular Client should win.

Given that the vast majority of litigation is conducted by lawyers on behalf of clients rather than by the clients themselves, we replicate this in our setup of the model. The Clients start off already hiring the lawyers to advocate the case on their behalf. Of course, once hired, the Client could provide the lawyers with no remuneration which would induce no effort from the lawyers.

The lawyers have two tasks: the solicitor’s tasks and the barrister’s tasks. The attorney is able to perform both tasks and has equal cost of effort for both tasks. The Solicitor can only perform the solicitor‘s tasks and the barrister may only perform the barrister‘s tasks. Their cost of effort j,k is also equal and equal to that of the attorney‘s with respect to that task. Let ei denote the effort of the lawyer for Client i performing task j = {s, b} (where s is the solicitor‘s tasks and b is the barrister‘s task). k denotes the effort level in the particular structure we are analysing; in s,sb this case, a for the Attorney system and sb for the Solicitor-Barrister system. For example, e1 denotes the effort level that the solicitor acting for client 1 exerts on the solicitor‘s tasks in the k s b Solicitor-Barrister system. Finally, we denote ei = ei + ei being the sum of the effort exert by lawyer i on the solicitor’s and barrister’s tasks. Where it is relevant, the superscript k is used to denote which structure of the profession we are analysing where k = {a, sb}

The lawyers are paid on a spec basis. This means they are only paid if the client is successful in the litigation. This type of arrangement is common practice and is provided for under legislation in many jurisdictions. For example, s323 of the Legal Profession Act 2004 (NSW) provides that a legal practitioner may conduct a matter on a conditional cost agreement where some or all of the legal costs are conditional on the successful outcome of the matter. s324 of the same Act provides that, in certain circumstances, the legal practitioner may apply an uplift fee to costs if there is a successful outcome.

Let Xi denote the payment that the lawyer acting for Client i receives if his/her client is successful in the litigation. We normalise the payment that the lawyer receives if the Client loses to zero (0).

As a result, Xi represents the extra amount (e.g. the uplift fee) that the lawyer receives conditional on the client winning the case.

We now define a linear probability function which describes the probability with which Client 1 wins the case as:  1 0 if k [(z1 + e1) − (z2 + e2)] ≤ − 2  P r(Client 1 wins) = 1 1 1 (1) k [(z1 + e1) − (z2 + e2)] + 2 if − 2 ≤ k [(z1 + e1) − (z2 + e2)] ≤ 2   1 1 if k [(z1 + e1) − (z2 + e2)] ≥ 2

6 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

This probability equation captures some important aspects of our modelling. Firstly, where the clients have equal justice values and the lawyers on both sides exert equal effort, both sides have equal probability of winning. Client 1’s probability of winning is increasing in his own justice value and his lawyer’s effort. Similarly, it is decreasing in the opponent’s justice value and the opponent’s lawyer’s effort. As this function is linear, it is necessary to restrict the range so that the probabilities fall between 0 and 1. Finally, note that it is entirely possible, in this model, for the Clients to choose a payment level of Xi = 0 and obtain effort level ei = 0 from their lawyers.

Now, we can describe the timeline of the model. Let us first focus on the Attorney System. The basic outline of the game is that the Clients choose the level of incentives after which the attorneys will choose their effort levels. More formally:

Stage 1 The clients simultaneously make decisions about the wage, Xi that they will offer to their attorney. Stage 2 The attorneys observe the wage offered to them by their clients and make decisions on s,a b,a the effort that he/she will exert: {ei , ei }. Stage 3 Payoffs are realised.

The model follows a similar structure in the Solicitor-Barrister system except that there is an additional element of the solicitor acting as an intermediary between the client and barrister. Our model follows the accepted structure that exists in all Common Law jurisdictions with Split profession systems (e.g. Australia, UK, Hong Kong). To some extent, this structure is enshrined in the regulations which govern barristers’ conduct. For example, Rule 21 of the NSW Barrister’s Rules provides that a barrister must accept any brief offered by a solicitor unless he/she is not available or the fee is unacceptable. However, Rule 24A provides that a barrister need not accept a brief if it comes directly from a Client. Furthermore, as in our model, it is the solicitor who is responsible for paying the barrister’s fee, not the client. General principles of privity of contract establish this proposition. Thus, the model structure in the Solicitor-Barrister system is:

Stage 1 The clients simultaneously make decisions about the wage, Xi that they will offer to their solicitor. Stage 2 The solicitors observe the wage offered to them by their clients and simultaneously makes s,sb decisions on the effort he/she will exert: ei and the wage contract, Yi, to offer to the barrister. Stage 3 The barristers observe the wage offered to them by the solicitor and the effort level exerted by the solicitor. The barrister then makes a decision on the effort level that he/she b,sb exerts: ei . Stage 4 Payoffs are realised.

Let the cost of effort functions be quadratic. Specifically, let the attorney’s cost of effort function a s,a b,a s,a 2 b,a 2 s s,sb s,sb 2 be C (ei , ei ) = (ei ) + (ei ) . Let the solicitor’s cost of effort function be C (ei ) = (ei ) b b,sb b,sb 2 and, similarly, let the barrister’s cost of effort function be C (ei ) = (ei ) . Finally, we assume that all agents are risk-neutral.

3.2 Equilibrium Characterisation

We are seeking a Subgame Perfect Nash Equilibrium in the following form:

s b s b  {X1,Y1, e1, e1}, {X2,Y2, e2, e2}

7 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

We can use simple backward induction techniques to solve this model. The basic intuition of the proof starts with the assumption that, if there is an equilibrium, it is interior. We show that there exists a unique interior equilibrium and find conditions under which it is, in fact, interior. Thus, if those conditions hold, there exists a unique interior equilibrium for this model. Our model does not consider the possibility of a corner solution equilibrium. However, intuitively, there can be no equilibrium where either lawyer exerts no effort (i.e. a corner solution) because there would always be an incentive for one of the lawyers to deviate and improve their chances of winning. An interested reader can refer to Appendix 1 [ for a more detailed explanation of the solution method.

We now turn to consider the interior solutions. Addressing first the Solicitor-Barrister system, start with the Barrister’s Problem. The Barrister observes the incentive provided by the solicitor and the solicitor’s effort level. Furthermore, the barrister anticipates the opposing barrister and solicitor’s effort levels. Thus, the barrister solves the following problem:   1 1 b 2 max k [(z1 + e1) − (z2 + e2)] + (Y1) − e (2) b 1 e1 2 2

The solicitor, anticipating the barrister’s actions, solved the following problem:

 1 1 max k [(z + e ) − (z + e )] + (X − Y ) − (es)2 (3) s 1 1 2 2 1 1 1 Y1,e1 2 2 s.t. (4)

 1 1 2 eb ∈ arg max k [(z + e ) − (z + e )] + Y − eb  1 1 1 2 2 2 1 2 1

We can then write down the problem that Client 1 faces:

 1 max k [(z1 + e1) − (z2 + e2)] + (π1 − X1) X1 2 s.t. s b e1 = e1 + e1 (5)  1 1   (es,Y ) ∈ arg max k [(z + e ) − (z + e )] + (X − Y ) − (es)2 1 1 1 1 2 2 2 1 1 2 1  1 1  2 eb ∈ arg max k [(z + e ) − (z + e )] + Y − eb  1 1 1 2 2 2 1 2 1

s b In the Attorney system, the process is similar except that the attorney chooses e1 and e1 simultaneously and does not need to choose a Y1. Thus, the Attorney solves the following problem:   1 1  s 2 b 2 max k [(z1 + e1) − (z2 + e2)] + (X1) − (e ) + e (6) s b 1 1 e1,e1 2 2 and the Client therefore faces the following problem:

 1 max k [(z1 + e1) − (z2 + e2)] + (π1 − X1) X1 2 s.t. (7) s b e1 = e1 + e1  1 1  2 es, eb  ∈ arg max k [(z + e ) − (z + e )] + X − (es)2 + eb  1 1 1 1 2 2 2 1 2 1 1

8 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Solving these problems yields the solution characterised in Proposition 1 below.

Proposition 1 There exist unique values k and k, where k > k > 0, such that this game has a unique interior equilibrium if and only if, k ∈ [k, k]. Furthermore, if k ∈ [k, k], then the interior equilibrium is the only equilibrium in the game.

1) That equilibrium takes the following form

 s∗ b∗ !  e1 , e1    X∗ ,Y ∗   1 1  (8)  s∗ b∗ !   e2 , e2    ∗ ∗ X2 ,Y2

2) In the Attorney system, it is specified by:

  3 3 3   2k(2π1+π2)−(z1−z2)− 2k 2k(2π1+π2)−(z1−z2)− 2k 2k(2π1+π2)−(z1−z2)− 2k 6 , 6 , 6k   3 3 3   (9) 2k(2π2+π1)−(z2−z1)− 2k 2k(2π2+π1)−(z2−z1)− 2k 2k(2π2+π1)−(z2−z1)− 2k 6 , 6 , 6k

Note that we do not need to specify a value for Yi in the Attorney System because it does not appear.

3) In the Solicitor-Barrister system, it is specified by:

  3 3   k(π1−π2)+(z1−z2)+ k k(2π2+π1)−2(z1−z2)− k 3 , 3   k2(2π +π )−k(z −z )− 3 k(2π +π )−2(z −z )− 3    1 2 1 2 2 2 1 1 2 k   3 , 3k  3 3 (10)   k(π −π )+(z −z )+ k(2π1+π2)−2(z2−z1)−    2 1 2 1 k [ k ]   3 , 3    2 3 3   k (2π2+π1)−k(z2−z1)− 2 k(2π1+π2)−2(z2−z1)− k 3 , 3k

Proof: See Appendix A

4 Results

We now discuss some results which follow from taking comparative statics derived from the equilibrium described in Proposition 1. Most importantly, we address some of the main trade- offs between the two systems that are highlighted by our model.

4.1 Incentives and Efficiency

Proposition 2 Total Effort is higher under a Fused Profession (i.e. attorney system) compared to a Split Profession system. That is: a s,a b,a sb s,sb b,sb e1 = e1 + e1 > e1 = e1 + e1 (11)

9 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Where:

2k (2π + π ) − (z − z ) − 3 ea = 1 2 1 2 2k (12) 1 3 k (2π + π ) − (z − z ) − 3 esb = 1 2 1 2 2k (13) 1 3

Proof: See Appendix B.1

An important implication arises from considering Propositions 2 and 3 jointly. Although the attorney always exerts more effort than the solicitor and the barrister jointly, the attorney does not always receive a higher incentive payment. Thus, there are cases where the attorney exerts more effort than the solicitor and the barrister but is paid less. This result implies that the attorney allocates effort between the two tasks more efficiently than the solicitor and the barrister.

This result is intuitive. In the Client’s payoff function, effort on the solicitor‘s task is a perfect substitute for effort on the barrister‘s task. Since the attorney allocates effort to both tasks simultaneously, he/she is able to frictionlessly substitute between the two tasks until the marginal cost of effort is equal to the marginal benefit. On the other hand, the transfer payment, Yi, given by the solicitor to the barrister, creates a friction to the substitution of effort across both tasks.

The reason that the incentive payment Yi creates friction is because of its role in the solicitor- barrister relationship. Examine the trade-off faced by the solicitor when choosing how to allocate effort on his task and how much incentive payment Yi to give to the barrister. If the solicitor provides no payment to the barrister and keeps all the client’s payment to himself (i.e. chooses

Yi = 0) the solicitor must exert all the effort. This is costly for the solicitor who faces convex costs. Thus, the solicitor could shift some of the effort cost to the barrister by inducing effort from the barrister. By increasing the incentive payment Yi, the solicitor gives up a share of the Client’s incentive payment in exchange for some effort from the barrister. Since the barristers cost of effort b,sb is convex at low levels of ei , the solicitor can induce more effort with relatively smaller incentive payments Yi.

However, as the solicitor attempts to induce more effort from the barrister, the solicitor must pay a larger incentive Yi to the barrister. This means that the solicitor must sacrifice a larger share of the client’s incentive and reduce his own share of the client’s incentive payment. On the other hand, increasing the payment Yi increases the probability that the client wins and increases the probability that the solicitor is paid. Thus, the solicitor can expend less effort whilst maintaining the same probability that the client will win and, consequently, that the solicitor earns the incentive payment. As such, the solicitor faces a trade-off. Paying a higher Yi to the barrister induces more effort from the barrister, increases the probability the solicitor will be paid Xi and allows the s,sb solicitor to exert a lower effort es . Conversely, paying a lower Yi means that the solicitor induces less effort from the barrister and the solicitor must, consequently, exert more effort to maintain the probability of winning.

It is this trade-off that causes the inefficiency in the allocation of effort. The solicitor’s trade- off and decision causes friction in the substitution of effort between the two tasks. In contrast, the s,a attorney is able to substitute frictionlessly between effort on the solicitor’s task (ei ) and effort b,a on the barrister’s task (ei ). In the attorney’s problem, effort on the solicitor’s task and effort

10 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita on the barrister’s task are perfect substitutes 3 However, the solicitor, whose decision effectively allocates effort between the two tasks, does not frictionlessly substitutes. For the solicitor, shifting effort between the solicitor’s tasks and the barrister’s tasks also entails a change in the allocation of the reward (i.e. the solicitor’s share of Xi given by Xi − Yi.). On the other hand, the attorney does change his share of the reward (Xi) when he shifts effort between the two tasks. As such, the change in the allocation of the reward causes the friction in the substitution of effort. For this reason, the allocation of effort in a Split Profession is less efficient than in a Fused Profession.

4.2 No Mere Mouthpiece - Independence of the Lawyers

Another important implication of the model arises when consider Proposition 3 below.

Proposition 3 3 For k(z1 − z2) > − 2 , the Attorney receives stronger incentives from the client as compared to the 3 solicitor. Where k(z1 − z2) < − 2 the client provides stronger incentives to the solicitor than to the attorney. (NB: the Client never provides incentives directly to the barrister)

Proof: See Appendix B.2

Proposition 3 establishes that the Attorney does not necessarily receive more incentives than the solicitor/barrister team. In some cases, the attorney receive smaller incentives than the solicitor and barrister. However, Proposition 2 establishes that the attorney always exerts more effort than the solicitor and barrister combined. This result raises an interesting issue: If the attorney is not receiving more incentives, why does he exert more effort? Proposition 4, below, provides one possible answer: The attorneys must be more responsive to the incentives than the solicitor and the barrister.

Proposition 4 b,sb The effect of the Client’s Incentives on the barrister‘s choice of effort, e1 and the solicitor’s choice s,sb s,a b,a of effort, e1 is strictly less than the attorney’s choice of effort on either task e1 , e1 . That is:

∂eb,sb ∂es,sb 1 ∂es,a ∂eb,a 1 = 1 = < 1 = 1 = k (14) ∂X1 ∂X1 2k ∂X1 ∂X1

Corollary 1 b,sb The effect of the Client’s payoff from wining on the barrister’s choice of effort, e1 and the s,sb solicitor’s choice of effort, e1 is strictly less than the attorney’s choice of effort on either task s,a b,a e1 , e1 . That is: ∂eb,sb ∂es,sb k ∂es,a ∂eb,a 2k 1 = 1 = < 1 = 1 = (15) ∂π1 ∂π1 3 ∂π1 ∂π1 3 Proof: See Appendix C

Proposition 4 confirms, in our interpretation, a phenomenon which is described by legal scholars as the independence of the lawyers from the client. Rule 18 of the NSW Barristers Rules state that a barrister should not be the “mere mouthpiece of the client”. The economic intuition behind this result is akin to a double marginalisation argument. In some sense, the Solicitor

3NB: This is a functional form assumption. However, this paper’s results do not depend on this assumption. All that is required is that the cost of effort for each task is identical for the attorney, the solicitor and the barrister.

11 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita and the Barrister are vertically separated in supplying legal services to the client. The Attorney is vertically integrated, supplying both types of legal services to the Client. Borrowing from the Industrial Organisation literature ((Spengler, 1950)), the vertically separated firms earn less profit than the vertically integrated firm because each firm in the structure adds it own price-cost margin when it supplies to the downstream firms. Thus, the monopoly problem is faced incurred twice. In this model, however, the result is not, strictly, one of a double monopoly problem. However, it may assist the reader to conceptualise the logic in this context.

In our model, the solicitor and the barrister are less affected by the incentives of the client because of the solicitor’s expropriation of some of the surplus in the relationship through the use of the incentives Yi. In both the Solicitor-Barrister system and the Attorney system, the client does not provide efficient incentives because he/she does not provide the full benefit of the lawyer’s effort as the compensation (i.e. Xi 6= πi). This occurs because the client expropriates some surplus from the relationship. This introduces an externality because the client receives some benefit at the expense of the lawyers. As a result, inefficient effort is exerted. However, in the Solicitor-Barrister system, this process is repeated again because the solicitor also expropriates some surplus from the relationship through the incentive Yi. As a result, the barrister is subjected to two expropriations of surplus – once from the client and once from the solicitor. As a result, the barrister receives lower incentives overall and is less affected by them compared to the attorney.

Furthermore, the trade-off that the solicitor faces (explained in Section 4.1 above) also means that the solicitor is less sensitive to the Client’s incentives and the Client’s payoff. The solicitor does not receive the full benefit of the incentive payment in that he must share it with the barrister in order for the barrister to exert effort. As such, the solicitor will be less influenced by the incentive payment.

This result is consistent with real world observation. For example, compare the US Attorney model with the Australian (NSW) Fused Profession model. Under the US model, attorneys are required to zealously advocate on behalf of their client because their primary duty is to their client, whilst NSW barrister and solicitors take a more modest approach to advocacy because their primary duty is to the Court. The practice of American attorneys is dominated by the primary duty of zealous advocacy (Lanctot, 1990). Indeed, in Walters v National Association of Radiation Survivors4, Rehquist CJ stated:

Under our adversary system, the role of counsel is not to make sure the truth is ascertained, but to advance his client’s cause by any ethical means. Within the limits of professional propriety, causing delay and sowing confusion not only are his right, but may be his duty. Walters v National Association of Radiation Survivors 473 U. S. 305 (1985) at 473 per Rehnquist CJ

Zitrin and Langford (1999) supports this view and presents anecdotal evidence of instances where the duty of zealous advocacy overrules other ethical considerations. Other authors support this view (see (Simon, 1988), (Richmond, 2002), (Luban, 1983)). On the other hand, the duty of the barrister is first and foremost to the Court. The preamble to the Barrister Rules 2001 (NSW) states that ‘ The administration of justice in New South Wales is best served by reserving the practice of law to officers of the Supreme Court who owe their paramount duty to the administration of justice’.

4473 U. S. 305 (1985) at 473

12 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

This is supported by the comments made by Chief Justice Mason in Giannarelli v Wraith:

The performance by counsel of his paramount duty to the court will require him to act in a variety of ways to the possible disadvantage of his client. Counsel must not mislead the court, cast unjustifiable aspersions on any party or witness or withhold documents and authorities which detract from his client’s case. And, if he notes an irregularity in the conduct of a criminal trial, he must take the point so that it can be remedied, instead of keeping the point up his sleeve and using it as a ground for appeal.

It is not that a barrister’s duty to the court creates such a conflict with his duty to his client that the dividing line between the two is unclear. The duty to the court is paramount and must be performed, even if the client gives instructions to the contrary. Giannarelli v Wraith (1988) 165 CLR 543 at 556-7 per Mason CJ

However, this does not mean that the barrister’s payoff should not be affected by the client’s stake in the litigation (πi) or the client’s incentive payment (Xi). If that were the case, the barristers would not have any incentive to advocate on behalf of their client and a significant incentive compatibility problem arises. In fact, the NSW Bar Rules contemplate the idea that the barrister (and solicitor for that matter) are entitled to an appropriate fee from the client in relation to that case and that a barrister may refuse a case if the fee offered is unreasonable. Thus, whilst the barrister must maintain independence from the client in the conduct of the case, the barrister need not be entirely financially independent of the client.

The difference in the two rules affecting the conduct of attorneys and barrister is reflected in the model through the effect of πi on the effort levels of the attorney compared to its effect on the effort of the solicitor and the barrister. πi represents the payoffs to client i’s case. The degree to which the attorney’s or the barrister’s choice of effort is affected by πi represents the degree to which they zealously advocate the client’s case. The model makes predictions which are consistent with the reality that the solicitor and the barrister will advocate the client case less zealously than the attorney. As a result, the lawyers under the Solicitor-Barrister system are more independent than lawyers under the Attorney system.

4.3 Justice

There is an extensive literature in the fields of sociology and legal philosophy which attempt to define justice. For example, Plato (60BC), in his most famous The Republic, considered that Justice is achieved by allocating to each person what ought to rightfully be theirs. In contrast, Rawls (1971) understood justice to mean distributive justice and stated that ‘each person is to have an equal right to the most extensive basic liberty compatible with a similar liberty for others’ (Rawls, 1971, p. 60) and ‘Social and economic qualities are to be arranged so that they are both reasonably expected to be to everyones advantage and attached to positions and offices open to all” (Rawls, 1971, p. 406). In contrast, Mill, basing his work off Bentham (1823), advocated a Utilitarian principle of justice which advocates that ‘that actions are right in proportion as they tend to promote happiness, wrong as they tend to produce the reverse of happiness.(Mill, 1863, p. 9).

It is not the place, nor purpose, of this paper to explore what is meant by justice. Many learned authors have explored this question extensively. For the purposes of this paper, we shall define justice as achieving the correct outcome in any given litigation. That is, if the client who has the

13 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita higher justice value wins the case, then justice has been achieved. Thus, in comparing between the two systems, we define a system as ‘better’ if the probability that the client with a higher justice value is more likely to win in that system compared to the other. Using this definition, we can now state Proposition 5.

Proposition 5 The system which better achieves justice depends on the correlation between which client has a higher justice value and which client has the higher payoff from winning. More formally:

1) If justice values and payoffs are positively correlated, that is z1 > z2 AND π1 > π2, then the Attorney system is better at achieving justice.

2 If justice values and payoffs are negatively correlated, that is z1 > z2 AND π1 < π2, then the Solicitor-Barrister system is better at achieving justice. Proof: See Appendix D

Note, of course, that one only needs to swap the subscripts to realise that the same results hold if we consider cases where z2 > z1. The intuition behind this result is quite simple but has some important implications. When justice values are positively correlated to payoffs, society prefers the client who has more to gain from winning to win the litigation. Since that client has a higher payoff to winning and a higher justice value, they will invest more in winning by providing a higher payment to their lawyers. A higher payment to the lawyers increases the effort the lawyers exert. This occurs in both a fused profession and a split profession structure. However, due to the leakages that arise in the split profession structure, the increase in effort as a result of the higher payment Xi is lower in the split profession structure. Where justice values and payoffs are positively correlated, it is socially optimal that the Client with the higher justice value (and thus, the higher payoff) should induce more effort from his lawyers. Thus, a structure of the legal profession which allows Client 1 to induce more effort from his lawyers is more likely to ensure that Client 1 is more likely to win. In this respect, a fused profession system is preferred.

The converse is true when justice values and payoffs are negatively correlated. In that case, the additional effort induced from the lawyers as a result of Client 2’s higher payoff distorts the probability function in favour of Client 2. To minimise this effect, the Solicitor-Barrister system ought to be used because, as was discussed in the Sections 4.1 and 4.2, the Solicitor and Barrister are less sensitive to the incentives provided by the Client. This means that Client 2 induces less effort from the lawyers than he would have under an Attorney-system. Accordingly, the effect of Client 2’s higher payment to the lawyers, and correspondingly the distorting effect of effort from those lawyers, is reduced under a Solicitor-barrister system as compared to the attorney system.

14 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

5 Application to Judicial and Legislative Law Making

The results presented in Section (4.3) can be applied to law and policy decisions. In particular, Proposition 5 explains that the optimal choice for the structure of the legal profession is linked to the relationship between the justice value and the client’s payoff. The relationship between the justice value and the client’s payoff is a factor that can be regulated by the development of the common law (in Common Law jurisdictions) and/or legislation (in both Common Law and Civil Law jurisdictions). Thus, Proposition 5 provides one important link between the structure of the legal profession in a jurisdiction and the types of law which exist in that jurisdiction. It is important to note here that we do not claim that the structure of the legal profession is the sole or even the primary determinant of the types of law a jurisdiction ought to pass. Rather, we provide one factor which ought to be taken into account by law and policy makers. Nevertheless, this paper provides an analysis of a previously-unexplored relationship between the structure of the legal profession and the type of law that ought to be passed.

5.1 The structure of the legal profession and law making

In interpreting the implications of Proposition 5, it is important to bear in mind the interpretation of each of the variables. Recall that zi represented the justice value of a client’s case; that is, how much does society think Client i ought to win adjudged by some objective criterion. The magnitude of zi is not particularly relevant. Rather, the relative magnitude of z1 and z2 is relevant since the client with the larger justice value ought to win. Secondly, πi refers to the client’s payoff from winning. More correctly, since we have normalised the value of losing a case to zero (0), πi represents the additional benefit accruing to the client as a result of winning the case. Thus, πi can be considered a measure of the damages or compensation from winning.

Proposition 5 states that the direction of correlation between the justice values and the client’s payoff is a determinant of which structure of the legal profession is more likely to achieve the correct outcome in litigation. If Client 1’s ought to win (i.e. Client 1’s justice value is higher than Client 2’s) and Client 1 has a high payoff from winning, then a Fused profession system is more likely to achieve the correct outcome. On the other hand, if Client 1 ought to win but has a smaller payoff from winning, then a Split profession will have a higher probability of ensuring the right outcome. The correlation between the justice values and the client’s payoff from winning is something that can be directly controlled from a law and policy perspective. A law-maker (be that a judge in the Common Law system or a Legislative body) can create to influence the direction of the correlation. For example, Parliament may decide to introduce laws which increase the compensation awarded where the defendant has been shown to act in gross negligence. Such legislation adjusts the correlation between the plaintiff’s payoff and the defendant’s payoff for any given pair of justice values. Thus, a law maker is able to control the correlation between the justice values and the clients’ payoffs. We refer to the correlation between the justice value and the client’s payoff as the type of law; a type of law with a positive correlation and a type of law with a negative correlation.

This result has a number of important applications. Firstly, using Proposition 5, when a jurisdiction is deciding which structure of the legal profession to adopt, a key consideration is the types of law that jurisdiction employs. When the correlation between justice values and payoffs is positive, then

15 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita a Fused Profession should be adopted. Where the reverse is true, a Split Profession is preferrable. Of course, it is important to note that, when considering which structure of the legal profession is more likely to achieve justice, the types of laws will vary. Most jurisdictions will enact laws with a positive correlation between the justice values and payoffs and other laws with a negative correlation. Thus, the decision of which structure to adopt is more complex than a comparison on a single set of laws. Rather, society must evaluate the relative importance of obtaining the correct outcome according to the different types of law. If society considers it more important to obtain the correct outcome in cases where the justice values and the payoffs are positively correlated, then it should adopt a Fused profession system.

An alternative way of looking at this is that, if the consequences of committing a Type I error significantly outweigh the costs of a a Type II error, then society should adopt a fused profession. These concepts are widely used in Hypothesis Testing and have been adapted to the Law and Economics literature in terms of litigation error (see Polinsky and Shavell (1989)). Type I and Type II errors can be easily converted to legally recognised concepts. Type I errors correspond to miscarriages of justice, where an innocent party is wrongly convicted of an offence or has damages awarded against them. Type II errors correspond to errors of impunity where a guilty party is found not guilty or a case is dismissed against the correct tortfeasor. In our model, a Type I error occurs when Client 1’s justice value is higher but Client 1 loses. A Type II error occurs when Client 1’s Justice value is lower but Client 1 wins. Again, the likelihood of avoiding a Type I or Type II error depends on the correlation between the justice values and the payoffs. The Attorney system is better at avoiding Type I error when there is positive correlation but the Solicitor-Barrister system is better when there is a negative correlation.

We can illustrate the nature of this trade-off in the context of capital punishment, otherwise known as the death penalty. Suppose that Client 1 is the defendant and Client 2 is the State prosecutor. Client 1 always has a higher payoff than Client 2 because Client 1’s payoff to winning is avoiding death. If Client 1 is guilty, then Client 1’s justice value is lower. Thus, there is a negative correlation between justice values and payoffs if Client 1 is guilty. If Client 1 is innocent, then Client 1’s justice value is higher and there is a positive correlation. Consider now Type I and Type II errors in this context. Type I errors occur when an innocent client is sentenced to death whilst a Type II error occurs when a guilty client is found not guilty. Under the death penalty, there is a strong imperative to avoid Type I errors whilst avoiding Type II errors is relatively more important without the death penalty. Thus, we expect that states with the death penalty will adopt a Fused legal profession whilst States without the death penalty will adopt a split profession system. This expectation is confirmed in practice with states such as and the US, with the death penalty, having a fused legal profession whilst states such as Australia and the UK having a Split Profession system in the absence of the death penalty.

6 Conclusion

This paper proposes a framework to answer the question: Under what circumstances does the Split Profession system produce better social outcomes than the Fused Profession system and vice-versa? We develop a theoretical model of the litigation process and use it to compare the outcomes arising under a Split Profession system against the outcomes arising under a Fused Profession system. We find that, although an attorney in a fused profession system will exert more

16 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita effort than the solicitor-barrister team in the split profession system, the attorney does not always receive a higher incentive payment. We also find a relationship between the relative justice values of the Clients, the relative payoffs of the Clients and which system is more likely to produce the correct outcome in litigation. Our findings relate the correlation between justice values and the payoffs with the structure of the legal profession. This has important implications for the choice of how to structure the legal profession. Furthermore, we show that the structure of the legal profession also has important implications for the choice of laws and policy making.

17 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Proof of Proposition 1

This Appendix provides the proof to proposition 1. First, we outline the logical steps in our proof and then proceed to illustrate the proof itself.

Begin with the assumption that there exists an interior equilibrium to the game. We characterise the interior equilibrium using backward induction to solve each agent’s maximisation problem. We find that only one interior equilibrium exists. We then show that, given the equilibrium is interior, there exists values k and k where k > k ≥ 0 such that k ∈ [k, k]. It is thus obvious that, if k ∈ k, k, then there exists a unique interior equilibrium. Finally, we show that there cannot exist any corner solution equilibria. This completes the proof of Proposition 1.

Note that we provide separate proofs for the Attorney Model and the Solicitor-Barrister Model. However, our proposition requires us to provide a unified set of k ∈ [k, k]. We use the intersection of the set of k in the Attorney Model and the set of k in the Solicitor Barrister Model ensuring that the resulting set of k holds in both proofs simultaneously.

Proof of Proposition 1 - Attorney Model

Suppose there exists an interior equilibrium. We now show how to characterise such those equilibria.

First, consider the Client’s problem. Each client faces the following problem:

 1 max k [(zi + ei) − (z−i + e−i)] + (πi − Xi) (16) Xi 2 s.t.

 1 1 h 2i es, eb ∈ arg max k [(z + e ) − (z + e )] + X − (es)2 + eb i i i i −i −i 2 i 2 i i

We can now solve for the Incentive Compatibility Constraint by solving the Attorney’s problem. For ease of notation, let us solve Client 1 and Attorney 1’s problem. Client 2 and Attorney 2’s problems are, of course, symmetrical to Client 1 and Attorney 1 respectively. Thus, the Attorney’s problem is:   1 1 h s 2 b 2i max k [(z1 + e1) − (z2 + e2)] + X1 − (e ) + e (17) s b 1 1 e1,e1 2 2

Taking First Order conditions, yields:

s kX1 − e1 = 0 (18) b kX1 − e1 = 0 (19)

Notice that (18) and (19) mean that:

s B e1 = e1 = kX1 (20)

Knowing Equation (20), we can substitute this back into the Client’s problem because the client correctly anticipates the effort level chosen by the attorney. Taking First Order conditions on the

18 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Client’s problem yields:

1 2k2π − k[(z + 4kX ) − (z + e )] − = 0 (21) 1 1 1 2 2 2

Solving Equation (21) yields:

2k2π − k(z − z ) + ke − 1 X = 1 1 2 2 2 (22) 1 4k2

s b We can also substituted Equation (22) into Equation (20) to find the optimal e1 and e1. Doing this yields: 2k2π − k(z − z ) + ke − 1 es = eb = 1 1 2 2 2 (23) 1 1 2k Using the fact that Client 1 and Client 2 are symmetric, we can state the following:

2k2π − k(z − z ) + ke − 1 X = 2 2 1 1 2 (24) 2 4k2 and 2k2π − k(z − z ) + ke − 1 es = eb = 2 2 1 1 2 (25) 2 2 2k We can now substitute Equation (25) into Equation (22) to solve for the Client 1’s optimal choice s b of X1 and into Equation (23) to find Attorney 1’s optimal choice of e1 and e1. Doing this, we have:

2k2(2π + π ) − k(z − z ) − 3 e∗ = 1 2 1 2 2 (26) 1 3k

By Symmetry: 2k2(2π + π ) − k(z − z ) − 3 e∗ = 2 1 2 1 2 (27) 2 3k Now substitute Equation (27 into Equation (22) and Equation (26) into Equation (24) to find the

Equilibrium values of X1 and X2. This yields:

2k2(2π + π ) − k(z − z ) − 3 X∗ = 1 2 1 2 2 (28) 1 6k2 and 2k2(2π + π ) − k(z − z ) − 3 X∗ = 2 1 2 1 2 (29) 2 6k2 Thus, we can characterise a Subgame Perfect Nash Equilibrium of the following form:

s b  s b  e1, e1,X1 , e2, e2,X2 (30)

From Equations (26), (28), (27), (29), we have the following equilibrium:

2k2(2π + π ) − k(z − z ) − 3 es∗ = eb∗ = 1 2 1 2 2 (31) 1 1 6k 2k2(2π + π ) − k(z − z ) − 3 es∗ = eb∗ = 2 1 2 1 2 (32) 2 2 6k 2k2(2π + π ) − k(z − z ) − 3 X∗ = 1 2 1 2 2 (33) 1 6k2 2k2(2π + π ) − k(z − z ) − 3 X∗ = 2 1 2 1 2 (34) 2 6k2

19 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

In order for these values to constitute an Interior Equilibrium, we require the following conditions:

s∗ b∗ e1 = e1 ≥ 0 (35) s∗ b∗ e2 = e2 ≥ 0 (36) ∗ π1 ≥ X1 (37) ∗ π2 ≥ X2 (38) 1 0 ≤ k [(z + e∗) − (z + e∗)] + ≤ 1 (39) 1 1 2 2 2

s∗ b∗ Beginning with e1 = e1 ≥ 0, we require:

2k2(2π + π ) − k(z − z ) − 3 1 2 1 2 2 ≥ 0 (40) 6k

2 3 This implies that: k > 0 and 2k (2π1 + π2) − k(z1 − z2) − 2 ≥ 0. Using the Quadratic formula, we see that: q 2 (z1 − z2) + (z1 − z2) + 12 (2π1 + π2) k ≥ (41) 4 (2π1 + π2) Notice that this is always positive since the denominator is positive and the numerator is always positive. Thus, we know that the binding condition is Equation (41) rather than the k > 0 condition.

Now turn to consider the X1 ≥ 0 condition. This implies:

2k2(2π + π ) − k(z − z ) − 3 1 2 1 2 2 ≥ 0 (42) 6k2

Notice that numerator is the same as in Equation (40) and the condition required on the denominator is k 6= 0. Thus, so long as Equation (41) holds, we ensure that X1 ≥ 0. Now, we need the second part of that condition required is that: π1 > X1. That is, we need:

2k2(2π + π ) − k(z − z ) − 3 π ≥ 1 2 1 2 2 (43) 1 6k2

Re-arranging this, we have:

3 2k2 (π − pi ) + k (z − z ) + ≥ 0 (44) 1 2 1 2 2

This means that we require the following condition:

q 2 − (z1 − z2) + (z1 − z2) − 24 (π1 − π2) k ≤ (45) 4 (π1 − π2)

Finally, we must address the condition:

1 0 ≤ k [(z + e∗) − (z + e∗)] + ≤ 1 (46) 1 1 2 2 2

Re-arrange this condition:

3 3 − ≤ 2k2 (π − π ) + k (z − z ) ≤ (47) 2 1 2 1 2 2

20 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

Since this is a quadratic equation in k, the maximum and minimum point of this function must be considered. Solving for the max/min-point yields:

z − z k = − 1 2 (48) 4 (π1 − π2)

3 3 This max/min point lies within the − 2 and 2 boundaries if:

(z − z )2 1 2 ≤ 12 (49) 4 (π1 − π2)

Notice that, even if the condition in Equation (49) does not hold, this means that there exists a gap in the possible set of k for which there is an interior equilibrium. However, that does not deny the existence of a non-empty set of k. Thus, it is possible to assume that Equation (49) holds without losing any generality or any qualitative results. For the rest of this proof, we assume that Equation (49) holds.

Now consider the boundaries points of the quadratic function given in Equation (47). Solving this yields one relevant condition:

q 2 − (z1 − z2) + (z1 − z2) − 24 (π1 − π2) k ≤ (50) 4 (π1 − π2)

Notice that Equation (50) is the same as Equation (45). Thus, we now have the following condition on k.:

q 2 q 2 (z1 − z2) + (z1 − z2) + 12 (2π1 + π2) − (z1 − z2) + (z1 − z2) − 24 (π1 − π2) ≤ k ≤ (51) 4 (2π1 + π2) 4 (π1 − π2)

Note that we have only solved these conditions from the perspective of Client 1. However, since Client 1 and Client 2 are symmetric in all respects, these conditions will be identical for Client 2.

So far, we have assumed that an interior equilibrium exists and showed there in fact exists only one interior equilibrium. We have also characterised that equilibrium and showed the conditions under which that equilibrium will be an interior equilibrium. That is, we showed there exists some values k > k ≥ 0 such that the equilibrium is interior if k ∈ [k, k]. Thus, it follows that if k ∈ [k, k] there exists a unique interior equilibrium for this game.

We now turn to show that there exists only one equilibrium in the game. Since we have just showed that there exists only one interior equilibrium, it is left to be shown that there does not exist any corner solutions. There are 5 possible corner solutions. The first two are cases where one or both Clients obtain no effort from their attorneys. That is:

e1 = 0, e2 = e2 (52)

e1 = e1, e2 = 0 (53)

e1 = 0, e2 = 0 (54)

Notice that the case where both Clients obtain zero effort (i.e. Equation (54) is not an equilibrium. Either Client could offer a positive payment to induce effort from their lawyers and instantly

21 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita increase their expected payoff. Similarly, in the case where one Client obtains zero effort from their lawyers is not an equilibrium. If one Client obtains zero effect from their lawyer, then the other Client could always reduce the incentive payment to their lawyers without significantly decreasing their expected payoff. Furthermore, the lawyers who exert zero effort would also do better by increasing their effort which would increase their expected payoffs.

Consider the second case of corner solutions where one set of lawyers exert sufficient effort to bring the probability that they will win to 1. In that case, the best that the other lawyers could do would be to lower their efforts to zero. However, we have just shown that there cannot exist an equilibrium where one set of lawyers exerts zero effort. Thus, there can be no set of equilibrium where the probability that one side will win is exactly equal to 1. Accordingly, there are no corner solutions. We can thus state Proposition 1 again:

There exist unique values k and k, where k > k > 0, such that this game has an interior equilibrium if and only if, k ∈ [k, k]. Furthermore, the game has only one interior equilibrium. Furthermore, if k ∈ [k, k], then the interior equilibrium is the only equilibrium in the game.

22 An Economic Model for Structures of the Legal Profession Ashley Cheng, Hodaka Morita

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