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									MICROECONOMIC ANALYSIS OF LAW
       September 19, 2006
MICROECONOMIC ANALYSIS OF LAW
       September 19, 2006
 MICROECONOMIC ANALYSIS OF LAW
        September 19, 2006
• Posted at:
• http://www.canlecon.org/
 MICROECONOMIC ANALYSIS OF LAW
        September 19, 2006
• http://www.cooter-ulen.com
• Answers to End of Chapter - Problems
                      BILATERAL AGENCY

                Bilateral Agency               Bilateral Contracts


                Principal Agency               Principal Agency
                                               Contracts




           Moral Hazard




                           Adverse Selection
Double
Moral Hazard
        BILATERAL AGENCY


The models that follow are simply
models.

The models simulate behaviour that
occurs across the legal system – not
what judges actually say or do in a
court.
                  BILATERAL AGENCY

.
                            Bilateral Agency




    Implicit Bilateral Agency                  Explicit Bilateral Agency
    Strategic                                  Strategic, relational
    Primarily market                           Primarily non-market
    Example – Cournot                          Example – Joint Venture
    Duopoly
    BILATERAL AGENCY - IMPLICIT

• Implicit Bilateral Agency
    »Relationship is strategic in nature
    • Examples: Duopoly – substitutes
    •            Duopoly – complements
   BILATERAL AGENCY - IMPLICIT


In many economic contexts implied
agencies arise.

These agencies involve non-legally
binding strategic interaction between
two or more agents.
  BILATERAL AGENCY - IMPLICIT
          COURNOT DUOPOLY



The most well known is the Cournot
duopoly, but there many other cases.
    BILATERAL AGENCY - IMPLICIT
              COURNOT DUOPOLY

• Agents operate economically similar
  firms – sole proprietorships:
          a = input of Agent 1
          1


          a = input of Agent 2
          2




         y = F(a ) = output of Agent 1
          1      1


         y = F(a ) = output of Agent 2
          2      2
     BILATERAL AGENCY - IMPLICIT
              COURNOT DUOPOLY

• Agents have “linear utility” in the
  profits they make. What does this
  mean?
           U(p ) = p = utility of Agent 1
              1    1


           U(p ) = p = utility of Agent 2
              2    2




• Agents are indifferent to risk - risk
  neutral
      BILATERAL AGENCY - IMPLICIT
              COURNOT DUOPOLY

• These agents have the following profit
  functions

 p (a ,a ) = (p-c)y = (1-y -y -c)y
  1   1   2       1        1       2           1


                    = y -y y -y y -cy
                       1   1   1       1   2       1




 p (a ,a ) = (p-c)y = (1-y -y -c)y
  2   1   2       2        1       2           2


                    = y -y y -y y -cy
                       2   2   1       2   2       2
           BILATERAL AGENCY - IMPLICIT
                       COURNOT DUOPOLY

• These agents act in their own self -
  interest (reaction curves)

 dp (a ,a )/da = 0
       1       1   2   1


 dp (a ,a )/da = 0
       2       1   2   2




 F (a ) – 2F(a )F (a ) - F (a )y -cF (a ) = 0
   1       1           1   1   1   1   1   2   1   1


 F (a ) – 2F(a )F (a ) - F (a )y -cF (a ) = 0
   2       2           2   2   1   2   2   2   2   2
BILATERAL AGENCY - IMPLICIT
      COURNOT DUOPOLY




       Set of Cost Minimizers


                        Set of Profit Maximizers
     BILATERAL AGENCY - IMPLICIT
    COURNOT DUOPOLY – NASH EQUILIBRIUM

• The principle or axiom of self-interest
is (reflected in reaction curves)

         F(a ) = (1/2)(1 - F(a ) - c)
             1                2


         F(a ) = (1/2)(1 - F(a ) - c)
             2                1
      BILATERAL AGENCY - IMPLICIT
      COURNOT DUOPOLY – NASH EQUILIBRIUM

• Equilibrium occurs
  where these “self-
  interested” actions
  intersect – Nash
  Equilibrium
a* = a* = F [(1/3)(1–c)]
  1    2   -1




• John Forbes Nash,
  1928 -
BILATERAL AGENCY - IMPLICIT
COURNOT DUOPOLY – NASH EQUILIBRIUM
  a2




       AGENT 1
       producing a1
       a1 = ½(1- a2 -c)



                     E[(1/3)(1-c), (1/3)(1-c)]


                                       AGENT 2 producing a2
                                       a2 = ½(1- a2 – c)
                                                              a1
        BILATERAL AGENCY - IMPLICIT
        COURNOT DUOPOLY – NASH EQUILIBRIUM

• If F (a) = a, the agents have the
  following Nash equilibrium:

 a* = a* = (1/3)(1 – c)
    1       2


 p* = p* = (1/9)(1 – c)(1 – c)
    1       2


 p* = 1 - (2/3)(1 – c) = (1/3)(1 + 2c)
           BILATERAL AGENCY - IMPLICIT
           COURNOT DUOPOLY – NASH EQUILIBRIUM

• If F (a) = a, the agents have the
  following iso-profit functions :

 p* = a -a a -a a -ca
       1      1       1   1   1   2   1


 a = - a - p /a + (1-c) - Agent 1
   2              1           1




 p* = a -a a -a a -ca
       2      2       2   2   2   1   2


 a = - a - p /a + (1-c) – Agent 2
   1              2           2
       BILATERAL AGENCY - IMPLICIT
       COURNOT DUOPOLY – NASH EQUILIBRIUM
         a2

Axes




          Iso-Profit Curve For Agent 2




                       E[1/3(1-c), 1/3(1-c)]
                            Iso-Profit Curve For Agent 1


                                                           a1
     BILATERAL AGENCY - IMPLICIT
     COURNOT DUOPOLY – NASH EQUILIBRIUM

• Professor Cooter both defines Nash
  equilibrium and distinguishes it from
  Pareto efficiency – (4th ed., 2004, c. 2.,
  VII, p. 41)
BILATERAL AGENCY - IMPLICIT




             Economic
             Measures
    BILATERAL AGENCY - IMPLICIT
               Market Efficiency

• Efficiency

    • An allocation of resources is
      efficient when no further increases
      to production can be made.
      BILATERAL AGENCY - IMPLICIT
                                    Market Efficiency


• Perfect Competition                        • Duopoly
                                                        [0,1]
    [0,1]
                                                        Consumer Demand
    Consumer Demand
                                                                P = 1-x
            P = 1-x
                                                                  [(2/3)(1-c), (1/3)(1+2c)]



                              [(1-c), c]
                      Producer Supply

                      [1,0]                                                       [1,0]
                        DECREASE in EFFICIENCY
    BILATERAL AGENCY - IMPLICIT
             Market Competitiveness



• Competitiveness

    • An allocation of resources is
      competitive when no further
      decreases to price can be made.
BILATERAL AGENCY - IMPLICIT
                  Market Competitiveness


• Perfect Competition                      • Duopoly
                                               [0,1]
    [0,1]
                                               Consumer Demand
    Consumer Demand
                                                       P = 1-x
            P = 1-x
                                                         [(2/3)(1-c), (1/3)(1+2c)]



                              [(1-c), c]
                      Producer Supply

                      [1,0]                                              [1,0]
                        DECREASE in Competitiveness
     BILATERAL AGENCY - IMPLICIT
           Market Optimality
• Professor Cooter explains Kaldor-Hicks
  “efficiency” – (4th ed., 2004, c. 2., IX, p.
  48)
• Mr. Justice Posner also uses the word
  “efficiency” in reference to “market
  optimality”
    BILATERAL AGENCY - IMPLICIT
               Market Optimality


• Pareto efficiency or Pareto optimality.
    • Maximizes social surplus making at
      least one individual better off,
      without making any other
      individual worse off.
    • An allocation of resources is
      Pareto optimal or Pareto efficient
      when no further improvements to
      social surplus can be made.
      BILATERAL AGENCY - IMPLICIT
                    Market Optimality

• Mr. Justice Posner offers a criticism of the
  Pareto criterion as being too narrow for policy
  formation. He uses the argument first raised by
  John Stuart Mill.
• “Every person should be entitled to the maximum
  liberty consistent with not infringing anyone else's
  liberty”.
• Because of the existence of interpersonal utility
  preferences, Mill's idea would contradict the
  strict application of the Pareto criterion to every
  case (6th ed., 2004, c. 1, pp. 12-13)
      BILATERAL AGENCY - IMPLICIT
                                    Market Optimality


• Perfect Competition                        • Duopoly
                                                        [0,1]
    [0,1]
                                                        Consumer Surplus
    Consumer Surplus
                                                                P = 1-x
            P = 1-x




                              [(1-c), c]
                      Producer Surplus

                      [1,0]                                                   [1,0]
                                                                Duopolists’
                        DECREASE in Social Surplus              Surplus
      BILATERAL AGENCY - IMPLICIT
            Market Optimality
• Kaldor-Hicks efficiency occurs when the
  economic value of social surplus is maximized.
• Under Kaldor-Hicks efficiency, a more optimal
  outcome can leave some people worse off.
• An outcome is more “optimal” or more “efficient”
  if those that are made better off could in theory
  compensate those that are made worse off.
     BILATERAL AGENCY - IMPLICIT
           Market Optimality
• As Mr. Justice Richard Posner quite rightly
  points out, the Kaldor-Hicks criterion – has
  limitations:
     • It does not answer the distributive issues.
     • Much of what economists call surplus is
       hypothetical
            » what consumers would pay for certain goods
            » not what is actually paid.
            » (6th ed., 2004, c. 1, p. 16)
      BILATERAL AGENCY - IMPLICIT
            Market Optimality
• .

                           Kaldor-Hicks
        Pareto             Criterion
        Criterion
   BILATERAL AGENCY - IMPLICIT
         Market Optimality
Recall that these models simulate
behaviour that occurs across the legal
system.

Exception: Antitrust cases. As a matter
of evidence, economic experts may
testify as to how social surplus is
effected by a merger or takeover
   BILATERAL AGENCY - IMPLICIT
         Market Optimality
Recently, the Federal Court of Appeal in
Canada ruled on the appropriateness of
using “social surplus” as a criterion for
evaluating a “friendly” merger between
ICG Propane and Superior Propane.
         BILATERAL AGENCY - IMPLICIT

.                        Implicit Bilateral Agency
                         Strategic
                         Primarily market
                         Example – Cournot
                         Duopoly




    Horizontal Implicit Agency                 Vertical Implicit Agency

    Example – Cournot Duopoly                  Example – Stackelberg
                                               Duopoly
      BILATERAL AGENCY - IMPLICIT

• Cournot Duopoly     • Stackelberg Duopoly




            AGENT 2           PRINCIPAL
AGENT 1




                                AGENT
    BILATERAL AGENCY - IMPLICIT
          STACKELBERG DUOPOLY

• The primary feature of the Stackelberg
  duopoly is that the “lead agent” takes
  into account not simply the existence
  of the rival agent (Cournot game) but as
  well its profit maximizing motivation.
     BILATERAL AGENCY - IMPLICIT
            STACKELBERG DUOPOLY

• The Stackelberg
  game lies behind
  many of the vertical
  relationships to be
  examined.

• Heinrich von
  Stackelberg, 1905-
  1946
    BILATERAL AGENCY - IMPLICIT
           STACKELBERG DUOPOLY

• Recall that the principle or axiom of
  self-interest for the Cournot duopoly
  was
           F(a ) = (1/2)(1 - F(a ) - c)
            1              2


           F(a ) = (1/2)(1 - F(a ) - c)
            2              1




 reflecting a “game” of simultaneous
 moves
      BILATERAL AGENCY - IMPLICIT
               STACKELBERG DUOPOLY

• The principle or axiom of self-interest for the
  Stackelberg duopoly is


  F(a ) = (1/2)(1 – [(1/2)(1 - F(a ) - c)] - c)
      1                                 1


  reflecting a “game” of sequential moves with the
  “lead agent” making the “first move” by optimizing
  its profits by taking the profit of the follower into
  account.
                 BILATERAL AGENCY - IMPLICIT
                                  STACKELBERG DUOPOLY

     • Duopoly                                                 • Stackelberg Duopoly
                                                                       Agent I       Isoprofit Curve of
[0,(1-c)]Agent I                                                                     Firm II



                                                                                        Isoprofit Curve
                                                                                        of Firm I
                                                               [0,(1/2)(1-c)]
[0,(1/2)(1-c)]
                    [(1/3)(1-c), (1/3)(1-c)]
                                                                                  [(1/2)(1-c), (1/4)(1-c)]



                                               Agent II                                           Agent II
                 [(1/2)(1-c), 0]
                                                  [(1-c), 0]                     [(1/2)(1-c), 0]
    BILATERAL AGENCY - IMPLICIT
          STACKELBERG DUOPOLY

• Equilibrium occurs, not where the “self-
  interested” actions of simultaneously
  moving players intersect, but where the
  profits of the “lead agent” are
  maximized:

         a* = F [(1/2)(1 – c)]
           1   -1


         a* = F [(1/4)(1 – c)]
           2   -1
             BILATERAL AGENCY - IMPLICIT

• Cournot Duopoly                          • Stackelberg Duopoly



                P = 1- a1 - a2                      P = 1- a1 - a2




 [(1/3)(1-c), 0] [(2/3)(1-c), 0]   [1,0]           [(1-c)/2,0]   [(3/4)(1-c), 0]   [1,0]
      BILATERAL AGENCY - IMPLICIT
                 STACKELBERG DUOPOLY


• Nash Equilibrium              • Nash Equilibrium
      • Simultaneous Solution         • Sequential Solution


      a1 = a2 = (1/3)(1-c)
                                      a1 = (1/2)(1-c)



                                      a2 = (1/4)(1-c)
                     BILATERAL AGENCY - IMPLICIT

        • Cournot Duopoly                          • Stackelberg Duopoly
                                                 [0,1]
[0,1]



                           P = 1- a1 - a2                               P = 1- a1 - a2
[0,(1/3)(1+2c)]
                                                 [0,(1/4)(1+ 3c)]




         [(1/3)(1-c), 0] [(2/3)(1-c), 0] [1,0]                      [(1-c)/2,0]   [(3/4)(1-c), 0]   [1,0]
     BILATERAL AGENCY - IMPLICIT
                 COURNOT DUOPOLY

• If F (a) = a, the agents have the
  following Nash equilibrium:
      a* = (1/2)(1 – c)
        1


      a* = (1/4)(1 – c)
        2


      p* = (1/8)(1 – c)(1 – c)
        1


      p* = (1/16)(1 – c)(1 – c)
        2


     p* = 1 - (3/4)(1 – c) = (1/4)(1 + 3c)
     BILATERAL AGENCY - IMPLICIT

• Cournot Benchmarks        • Stackelberg Benchmarks
     • Efficiency                • Efficiency
     a1 + a2 = (2/3)(1-c)        a1 + a2 = (3/4)(1-c)

     • Competitiveness
                                 • Competitiveness
     p = (1/3)(1 + 2c)
                                 p = (1/4)(1 + 3c)
     • Producers Surplus
                                 • Producers Surplus
     PS = (2/9)(1-c)(1-c)
                                 PS = (3/16)(1-c)(1-c)
     • Social Surplus
                                 • Social Surplus
     SS = (4/9)(1-c)(1-c)
                                 SS = (15/32)(1-c)(1-c)
BILATERAL AGENCY - IMPLICIT




             Collusive
             Duopoly
   BILATERAL AGENCY - IMPLICIT
            COLLUSIVE DUOPOLY




•With no property rules - can contracts still
 exist?
       BILATERAL AGENCY - IMPLICIT
                  COLLUSIVE DUOPOLY
        a2

Axes




         New Iso-Profit Curve For Firm Y


                         New Nash Equilibrium
                       New Iso-Profit Curve For Firm X


                                                         a1
   BILATERAL AGENCY - IMPLICIT
            COLLUSIVE DUOPOLY




•Yes. The collusive contract is more optimal
 for both parties, but is unstable. Either party
 has a “short-term” incentive to “defect” to the
 Nash equilibrium
BILATERAL AGENCY - IMPLICIT
                                COLLUSIVE DUOPOLY


• Perfect Competition                      • Collusive Duopoly
                                                [0,1]
    [0,1]
                                                Consumer Demand
    Consumer Demand
                                                        P = 1-x
            P = 1-x
                                                          [(1/2)(1-c), (1/2)(1+c)]



                              [(1-c), c]
                      Producer Supply

                      [1,0]                                                [1,0]
                        BIGGER DECREASE in
                        EFFICIENCY
BILATERAL AGENCY - IMPLICIT
                                COLLUSIVE DUOPOLY



• Perfect Competition                      • Collusive Duopoly
                                                [0,1]
    [0,1]
                                                Consumer Demand
    Consumer Demand
                                                        P = 1-x
            P = 1-x
                                                          [(1/2)(1-c), (1/2)(1+c)]]



                              [(1-c), c]
                      Producer Supply

                      [1,0]                                                [1,0]
                        BIGGER DECREASE in
                        Competitiveness
BILATERAL AGENCY - IMPLICIT
                                COLLUSIVE DUOPOLY


• Perfect Competition                      • Collusive Duopoly
                                                [0,1]
    [0,1]
                                                Consumer Surplus
    Consumer Surplus
                                                        P = 1-x
            P = 1-x




                              [(1-c), c]
                      Producer Surplus                     Duopolists’ Surplus

                      [1,0]                                              [1,0]

                  BIGGER DECREASE in Social Surplus
BILATERAL AGENCY - IMPLICIT
       COLLUSIVE DUOPOLY




            OPTIMAL
            LAW
       BILATERAL AGENCY - IMPLICIT
             COLLUSIVE DUOPOLY

• Recall Smith’s argument that “optimal”
  rules should make society better off
  economically

 The “central problem” for “lawmakers” is to
 maximize social surplus

• Which alternative maximizes social
  surplus?
       BILATERAL AGENCY - IMPLICIT
            COLLUSIVE DUOPOLY

• Outcome 1: A law or a rule that would
  “prohibit” collusive contracts!
• Outcome 2: A law or a rule here that
  would “ignore” collusive contracts, but
  choose not to enforce them should they be
  breached!
• Outcome 3: A law or a rule that would
  “enforce” collusive contracts!
       BILATERAL AGENCY - IMPLICIT
               COLLUSIVE DUOPOLY

• The “Legal” Problem
    • “Hypothetical” social planner – Dictator
    •                               - Judge
• Maximize social surplus
    • Subject to the requirement that Agent 1 maximizes
      its profits (Agent 1 is rational)
    • Subject to the requirement that Agent 2 maximizes
      its profits (Agent 1 is rational)
     BILATERAL AGENCY - IMPLICIT
          COLLUSIVE DUOPOLY

.
              Social Planner




    AGENT 1                    AGENT 2
     BILATERAL AGENCY - IMPLICIT
             COLLUSIVE DUOPOLY

• Note the “Stackelberg” nature of the “legal
  problem”?
• Coincidence or are there any worthwhile
  analogies?
       BILATERAL AGENCY - IMPLICIT
                COLLUSIVE DUOPOLY

• Maximize SS
    • Subject to F (a ) – 2F(a )F (a ) - F (a )y -cF (a ) = 0
                    1   1       1   1   1   1   1   2   1   1


    • Subject to F (a ) – 2F(a )F (a ) - F (a )y -cF (a ) = 0
                    2   2       2   2   1   2   2   2   2   2




• Simple case F(a) = a:
• Maximize SS
    • Subject to 1 – 2F(a ) - F(a ) – c = 0
                            1       2


    • Subject to 1 – 2F(a ) - F(a ) – c = 0
                            2       1
        BILATERAL AGENCY - IMPLICIT
                 COLLUSIVE DUOPOLY

• L = SS(a ,a ) + l (1 – 2F(a ) - F(a ) – c)
             1   2           1           1   2


     + l (1 – 2F(a ) - F(a ) – c)
         2                   2       1




• The “legal problem” adds these first order
  conditions to the “duopoly problem”:
          dL(a ,a )/da = 0
                     1   2       1


          dL(a ,a )/da = 0
                     1   2       2
     BILATERAL AGENCY - IMPLICIT
                 COLLUSIVE DUOPOLY

• Note the “self-interest” of each duopolistic
  agent still “applies” or is “binding”:
          dp (a ,a )/da = 0
             1    1       2   1


          dp (a ,a )/da = 0
             2    1       2   2




• So this means:
              l ≠0    1


              l ≠0    2
    BILATERAL AGENCY - IMPLICIT
            COLLUSIVE DUOPOLY

• Outcome 2: A law or a rule here that
  would “ignore” collusive contracts, but
  choose not to enforce them should they be
  breached!
• Best satisfies the “legal problem”
• This closely approximates the common
  law as it existed in Canada until 1889
     BILATERAL AGENCY - IMPLICIT
              COLLUSIVE DUOPOLY

• What happened?
• In 1889 after complaints about a Toronto
  coal cartel, fire insurance cartel, etc, the
  government “criminalized” collusive
  agreements – 1889 to 1990
     BILATERAL AGENCY - IMPLICIT
              COLLUSIVE DUOPOLY

• Some argue that
  Canada’s first
  antitrust legislation
  was designed to ward
  off the effects of
  monopoly due to Sir
  John A. Macdonald’s
  National Policy
       BILATERAL AGENCY - IMPLICIT
              COLLUSIVE DUOPOLY


• After 1990 – collusive agreements were
  decriminalized and are now subject to an
  elaborate administrative process
  supervised by the Competition Bureau
       BILATERAL AGENCY - IMPLICIT
                 COLLUSIVE DUOPOLY


• Closer in some cases to
    • Outcome 1: A law or a rule that would “prohibit”
      collusive contracts!
• This would suggest a “sub-optimal” choice
  by the “social planner”. Why?
     BILATERAL AGENCY - IMPLICIT
             COLLUSIVE DUOPOLY

• There is another key issue here.
• Note that a rule that does not enforce the
  “collusive contract” is a “complement” to
  the Prisoner’s dilemna
• A form of “strategic complementarity”
     BILATERAL AGENCY - IMPLICIT
               COLLUSIVE DUOPOLY

• Return to this specific issue under “Firms”
• This issue and related “antitrust” issues
  are studied in
     • ECO310Y5
     • Industrial Organization and Public Policy
BILATERAL AGENCY - IMPLICIT




            DEFECTION
    BILATERAL AGENCY - IMPLICIT
             PRISONERS DILEMNA

• Cooter explains the Prisoner's dilemna –
  (4th ed., 2004, c. 2., VII, p. 39)
     BILATERAL AGENCY - IMPLICIT
                PRISONERS DILEMNA

• Outcome 2: A law or a rule here that would
  “ignore” collusive contracts, but choose
  not to enforce them should they be
  breached!
     • Outcome 2 involves the operation of the Nash
       equilibrium that motivates a Prisoners dilemna
       outcome
     • So a law, rule or policy, as was the common law,
       that does not enforce the contract “complements”
       the Prisoner dilemna outcome
    BILATERAL AGENCY - IMPLICIT
               PRISONERS DILEMNA

• What exactly happens?
    • Agent 1 decides to “defect” from the agreed upon
      quota by increasing its profits at the “monopoly”
      price that resulted when the agents decided to
      collude:
          » Output of each agent = (1/4)(1-c)
          » Market Price         = (1/2)(1 + c)
          » Adjusted Output of Agent 1
                                 = (3/8)(1 - c)
       BILATERAL AGENCY - IMPLICIT
                   PRISONERS DILEMNA
        a2

Axes




         Collusive Iso-Profit Curve For Firm Y




                        Collusive New Iso-Profit Curve
                        For Firm X

                                                         a1
     BILATERAL AGENCY - IMPLICIT
                PRISONERS DILEMNA

• In the first “round” Agent 1 has increased
  its production by 50%
     • Agent 2 “reacts” to the defection from the quota by
       expanding its production to meet the falling market
       price:
           » Total Output of Agents     = (5/8)(1-c)
           » Market Price falls to      = (1/8)(3 – 5c)
           » Adjusted Output of Agent 2
                                        = (5/16)(1 - c)
     BILATERAL AGENCY - IMPLICIT
                PRISONERS DILEMNA

• In the second “round” Agent 2 has
  increased its production by 25%

    • Agent 1 “reacts” to Agent 2 expanding its
      production to meet the falling market price:

           » Re-adjusted Output of Agent 1
                  = (11/32)(1 - c)
     BILATERAL AGENCY - IMPLICIT
             PRISONERS DILEMNA

• In each successive “round” the agents
  readjust their outputs in response to each
  other until the original production Nash
  equilibrium is reached

 Output of each agent = (1/3)(1-c)
       BILATERAL AGENCY - IMPLICIT
                 PRISONERS DILEMNA
        a2

Axes




         Iso-Profit Curve For Agent 2




                      E[1/3(1-c), 1/3(1-c)]
                           Iso-Profit Curve For Agent 1


                                                          a1
   BILATERAL AGENCY - IMPLICIT
           COLLUSIVE DUOPOLY




Is there a way to make the
collusive contract more stable?
   BILATERAL AGENCY - IMPLICIT
           COLLUSIVE DUOPOLY




What happens if Agent 1 cannot observe
the effort of Agent 2?
   BILATERAL AGENCY - IMPLICIT
           COLLUSIVE DUOPOLY




What happens if Agent 1 does not know
the costs of Agent 2?
   BILATERAL AGENCY - IMPLICIT
          STACKELBERG DUOPOLY




What happens in the Stackelberg duopoly?
Does either the leader or the follower defect?
         BILATERAL AGENCY - EXPLICIT

.                       Explicit Bilateral Agency
                        Strategic
                        Primarily market




    Imposed Explicit Agency                  Voluntary Explicit Agency

    Example – No Fault                       Example – Negotiated
    Insurance Among Automobile               Contract
    Drivers
    BILATERAL AGENCY - EXPLICIT

• Explicit Bilateral Agency
    » Relationship is both strategic and has some legal
          significance
    » Imposed – A law “imposes” a relationship onto
                parties
    » Examples: Parent – child
                Car owner – accident victim
    BILATERAL AGENCY - EXPLICIT

• Explicit Bilateral Agency

    » Voluntary – The parties “choose” their relationship
    » Examples:
    • Business partnerships
    • Landlord – Tenant leases
    • Buy – Sell agreements
    BILATERAL AGENCY - EXPLICIT

• Restraints and incentives to the work
  ethic

• Effect of risk on contracts - where do
  agency costs originate?
         BILATERAL AGENCY - EXPLICIT

.                           Explicit Bilateral Agency
                            Strategic, Relational
                            Primarily Non-market




    Horizontal Explicit Agency                   Vertical Explicit Agency

    Example – Partnership                        Example – Employment
    Contract                                     Contract
BILATERAL AGENCY - EXPLICIT
                  Horizontal Contract




                 Promise of
                Agent 2
      AGENT 1                 AGENT 2

                 Promise
                of Agent 1
    BILATERAL AGENCY - EXPLICIT

• Horizontal Contracts

    » Examples:Two partners in a firm
               Two joint property
               owners
               Spouses
  BILATERAL AGENCY - EXPLICIT
            Horizontal Contract



   Explicit agencies arise when rules
align the "self-interest" of the agents to
the "common" objective of the agency.

   The chief feature is a "rule of law"
that binds the agents' self-interest to
the common objective.
    BILATERAL AGENCY - EXPLICIT
          Horizontal Contract
• Each agent exchanges the performance
  or execution of a promise for a
  payment.
• Each agent cannot observe the effort or
  action applied by the other party.
• This means neither agent cannot know
  in advance whether or not the contract
  will be performed. (Double Moral
  Hazard)
    BILATERAL AGENCY - EXPLICIT
              Horizontal Contract

• Different “sharing rules” include:
     • rights to residual profits
     • Profit - sharing
     • sharing the return to an investment
     • performance pay
     • fixed wage and
     • piece rate.
    BILATERAL AGENCY - EXPLICIT
              Horizontal Contract

• Agents decide to enter into a
  “collusive” contract with a view to:

    • Overcoming the Prisoners dilemna
    • Overcoming the inability to observe
      each others effort
BILATERAL AGENCY - EXPLICIT
          Horizontal Contract

• Overcoming the inability to enforce a
  broken contract because
      »No courts or judges are available
      »The available courts cannot
       observe the efforts and do not
       have evidentiary means to
       overcome this
      »The judges accept bribes from
       parties before them
      »The contracts are illegal
     BILATERAL AGENCY - EXPLICIT
              Horizontal Contract

.
                Social
                Planner



    AGENT 1                   AGENT 2
    BILATERAL AGENCY - EXPLICIT
              Horizontal Contract

• Agents enter into a partnership or joint
  venture called a “bilateral contract”:

         a = input of Agent 1
          1


         a = input of Agent 2
          2


         y = F(a ,a ) = joint output of
                 1   2


         Agents 1 and 2
    BILATERAL AGENCY - EXPLICIT
                 Horizontal Contract

• As before, agents have “linear utility”
  in the profits they make.

         U(p ) = p = utility of Agent 1
             1       1


         U(p ) = p = utility of Agent 2
             2       2
    BILATERAL AGENCY - EXPLICIT
              Horizontal Contract

• Mr. Justice Richard Posner argues on
  the basis that man is a rational utility
  maximizer in all areas of life, including
  legal matters (6th ed., 2004, c. 1, p. 4)

• How does Posner defend this? In terms
  of group behaviour – not individual
  aberrations. (6th ed., 2004, c. 1, p. 18)
     BILATERAL AGENCY - EXPLICIT
                  Horizontal Contract

• These agents have the following joint
  profit function:
           p(a ,a ) = py - ca - ca
              1    2               1       2




• For simplicity, let p = c = 1
         p(a ,a ) = F(a ,a )- a - a
              1    2       1   2       1       2
    BILATERAL AGENCY - EXPLICIT
              Horizontal Contract

• Agents are indifferent to risk - risk
  neutral

• The agents agree to adopt a sharing
  rule, or alternatively, the social planner
  agrees to “impose” an optimal sharing
  rule on the agents.
BILATERAL AGENCY - EXPLICIT
        Horizontal Contract

      LEGAL ANALYSIS   ECONOMIC ANALYSIS
      Agent 1          Agent 1
                       INCENTIVE
      PROMISED         COMPATIBILITY
      PERFORMANCE 1    CONSTRAINT 1

      Agent 2          Agent 2

                       INCENTIVE
      PROMISED         COMPATIBILITY
      PERFORMANCE 2    CONSTRAINT 2
    BILATERAL AGENCY - EXPLICIT
                 Horizontal Contract

• The principle or axiom of self-interest
applies as each agent is “rational”:

    dp(a ,a )/da = aF (a ,a )– 1 = 0
         1   2     1      1   1       2


    dp(a ,a )/da = (1-a)F (a ,a )– 1 = 0
         1   2     2              2       1   2
     BILATERAL AGENCY - EXPLICIT
              Horizontal Contract


Each “incentive compatibility constraint”
is binding because the first order
conditions hold due to the “self-interest”
of each “rational” agent:

    l1(aF (a ,a )– 1) > 0
          1   1       2


    l2((1-a)F (a ,a ) – 1) > 0
                  2       1   2
     BILATERAL AGENCY - EXPLICIT
           Horizontal Contract



Each “shadow price”, l1 > 0 and l2 > 0,
reflects the value to each agent of
contractual performance.
     BILATERAL AGENCY - EXPLICIT
           Horizontal Contract
On the other hand, “individual rationality
constraints” are not binding. No “direct”
principal makes payments to the agents.

Nor are any restrictions or constraints
placed on the agents’ abilities to make
transfer payments to each other.

So 1 = 0 and 2 = 0
EXPRESS BILATERAL AGENCY

 LEGAL ANALYSIS   ECONOMIC ANALYSIS
 Agent 1          INCENTIVE
 PROMISED         COMPATIBILITY
 PERFORMANCE 1    CONSTRAINT 1
 Agent 2          INCENTIVE
 PROMISED         COMPATIBILITY
 PERFORMANCE 2    CONSTRAINT 2
 Agent 1          PARTICIPATION
 PROMISED         CONSTRAINT 1
 PAYMENT 1

 Agent 2          PARTICIPATION
 PROMISED         CONSTRAINT 2
 PAYMENT 2
       BILATERAL AGENCY - EXPLICIT
                       Horizontal Contract

• As before, the “social planner” acts to
  maximize social surplus so as to
  optimize the applicable legal rule:

  L(a ,a ) = aF(a ,a )+ (1-a)F(a ,a ) - a - a
       1   2            1   2                1       2       1   2


+ l (aF (a ,a ) – 1) + l ((1-a)F (a ,a )– 1)
   1       1   1   2            2       2        1       2




  where a represent Agent 1’s share and
  (1-a) represents Agent 2’s share
    BILATERAL AGENCY - EXPLICIT
             Horizontal Contract



• The “legal problem” requires these
  first order conditions:

         dL(a ,a )/da = 0
             1   2   1


         dL(a ,a )/da = 0
             1   2   2


         dL(a ,a )/da = 0
             1   2
BILATERAL AGENCY - EXPLICIT
       Horizontal Contract

  Under the restrictive assumption of
  linear costs, the first order conditions
  are:
   dL/da1 = F1 - 1 + l1aF11 + l2(1-a)F12 = 0
   dL/da2 = F2 - 1 + l1aF12 + l2(1-a)F22 = 0
   dL/da = l1F1 - l2F2 = 0
BILATERAL AGENCY - EXPLICIT
           Horizontal Contract


The solution to the first order conditions
generates the optimal “sharing rule”,
which satisfies:

         a/(1- a) = (F22/F11)^1/4

Neary, Hugh and Winter, Ralph, “Output Shares in Bilateral Agency Contracts”,
(1995), 66 Journal of Economic Theory 609-614
    BILATERAL AGENCY - EXPLICIT
                Horizontal Contract

• Exercise:
    • What conclusions change, if any,
      when the agents are price takers,
      price searchers and have costs?

         p(a ,a ) = py - ca - ca
            1    2            1       2
 PRINCIPAL - AGENCY

 “SUPER”
 Principal             Its “problem” is to maximize social
             surplus




              Principal


promise
                   payment


    AGENT
    BILATERAL AGENCY - EXPLICIT

• Vertical Agency
    » Two parties agree on a ranking and
         order of conduct
              Principal – first mover
              Agent – second mover
    » Examples: Landlord and tenant – (residential)
               Employer - employee
               Buyer – seller
               Client - lawyer
BILATERAL AGENCY - EXPLICIT
    Vertical Contract – (Principal – Agency)

• “Principal Agency” Exchange
    • The “principal” makes an exchange of a
      “payment” to an “agent” in exchange for the
      “agent” performing or executing a
      “promise” for the “principal”
    • Again – note the “Stackelberg”
      structure of the agency
BILATERAL AGENCY - EXPLICIT




            Single Moral
             Hazard - I
BILATERAL AGENCY - EXPLICIT
    Vertical Contract – (Principal – Agency)

Single Moral Hazard

The principal cannot observe the effort or
action applied by the agent.

This means the principal cannot know in
advance whether or not the contract will
be performed by the agent.
               BILATERAL AGENCY - EXPLICIT
                   Vertical Contract – (Principal – Agency)

• Professor Cooter illustrates some
  cases of how moral hazards emerge in
  agency relationships:
•    A used car seller knows more about
  the quirks of his car than the buyer
•    A bank presents a “standard”
  deposit agreement to the customer
• (4th ed., 2004, c. 2., IX, p. 47)
    BILATERAL AGENCY - EXPLICIT
          Vertical Contract – (Principal – Agency)

• Parties enter into a “principal-agency”
  contract:

    a = 0 = input of Principal
      1


    a = input of Agent 2
      2


    y = F(0,a ) = output of the agency
                   2
    BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)

• This agency has the following profit
  function:
          p(0,a ) = pay - ca
                   2                  2




• For simplicity, let p = c = 1
         p(0,a ) = aF(0,a ) - a
                   2          2           2


         p(a ) = aF(a ) - a
               2          2       2
     BILATERAL AGENCY - EXPLICIT
             Vertical Contract – (Principal – Agency)


 p(a ) = aF(a ) - a
     2               2       2




 F’(a ) = dF/da > 0
         2               2           Production
                                     Concavity = Marginal
                                     Diminishing Returns

 F’’(a ) = d(dF/da )/da < 0
             2               2   2
    BILATERAL AGENCY - EXPLICIT
      Vertical Contract – (Principal – Agency)

• Principal is indifferent to risk - risk
  neutral
• The agent is risk averse. Why?
• Most parties in the real world are risk
  averse. Principals are risk averse. In
  relative terms, agents are even more
  risk averse
     BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)



 dU(W - a )/d(W - a ) > 0
        2       2       2       2       Risk Averse Utility of
                                        Agent



 d[dU(W - a )/d(W - a )]\d(W - a ) < 0
            2       2       2       2      2    2
    BILATERAL AGENCY - EXPLICIT
      Vertical Contract – (Principal – Agency)

• Mr. Justice Richard Posner uses the
  existence of insurance markets and the
  higher return on stocks over bonds as
  empirical evidence of widespread risk
  aversion (6th ed., 2004, c. 1, p. 11)
  BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)



U(F)
                                    A “perfectly competitive”
                                    risk neutral Principal
                                    contracts a “complete”
                                    contract with the “risk
                                    averse” agent


                                    Contract Equilibrium
          E                         Point

                                    The parties are paid in
                                    “output” shares


                                                 F=Output
     BILATERAL AGENCY - EXPLICIT
           Vertical Contract – (Principal – Agency)

• Professor Cooter explains that the
  utility function of a “risk-averse” agent
  is “concave downwards” - reflecting
  marginal diminishing utility of income
  (or output shares). (4th ed., 2004, c. 2., X, p. 51)
        BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)



The “participation constraint” of the Agent
is binding

2 (T + (1-a)F(a )- a ) > 0
    2               2    2
         BILATERAL AGENCY - EXPLICIT
          Vertical Contract – (Principal – Agency)



W = T + (1-a)F(a )
     2      2             2 “linear contract”
W = T + (1-a)F
     2      2


T = “insured” payment under the contract
 2


(1-a)F = “performance” payment under the
contract
     BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency

• If the Agent maximizes its utility
           dU(W - a )/da = 0
                     2   2       2


           dU(T + (1-a)F(a ) - a )/da = 0
                 2                       2   2   2


             (0 + (1-a)F (a ) - 1) = 0
                                     2   2




                 (1-a)F (a )– 1 = 0
                             2       2
    BILATERAL AGENCY - EXPLICIT
      Vertical Contract – (Principal – Agency

• So the Agent’s “incentive compatibility
  constraint” is also binding
         l2((1-a)F (a )– 1) = 0
                     2   2


         l2 > 0
         2 > 0
BILATERAL AGENCY - EXPLICIT




            Single Moral
             Hazard - II
BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)



      LEGAL ANALYSIS        ECONOMIC ANALYSIS
     Agent 1                INCENTIVE
     PROMISED               COMPATIBILITY
     PERFORMANCE 1          CONSTRAINT 1
     Agent 2                INCENTIVE
     PROMISED               COMPATIBILITY
     PERFORMANCE 2          CONSTRAINT 2
     Agent 1                PARTICIPATION
     PROMISED               CONSTRAINT 1
     PAYMENT 1

      Agent 2               PARTICIPATION
      PROMISED              CONSTRAINT 2
      PAYMENT 2
      BILATERAL AGENCY - EXPLICIT
              Vertical Contract – (Principal – Agency)

• The “social planner” acts to maximize
  social surplus so as to optimize the
  applicable legal rule:

 L(a ,a ) = aF(a ) - a + l ((1-a)F (a )– 1) +
      1       2          2       2       2     2   2


  (T + (1-a)F(a )– a )
  2       2                  2       2
BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)



      LEGAL ANALYSIS        ECONOMIC ANALYSIS

     Principal              Agent
              PROMISED
             PAYMENT        PARTICIPATION
                            CONSTRAINT
      Agent

                     INCENTIVE
         PROMISED    COMPATIBILITY
         PERFORMANCE CONSTRAINT
    BILATERAL AGENCY - EXPLICIT
      Vertical Contract – (Principal – Agency)



• The “legal problem” requires these
  first order conditions:

          dL(a ,a )/da = 0
                1   2   2


          dL(a ,a )/da = 0
                1   2
BILATERAL AGENCY - EXPLICIT
        Vertical Contract – (Principal – Agency)


  Under the restrictive assumption of
  linear costs, the first order condition
  for sharing is:
              dL/da = F - l2F2 - 2F = 0

                           F = 2F + l2F1
                           1 = 2 + l2F2/F
BILATERAL AGENCY - EXPLICIT
    Vertical Contract – (Principal – Agency)




The optimal “sharing rule”, satisfies:

    1 = 2 + l2F2/F
  BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)



U(F)
                                    There is a “third”
                                    constraint” in the
                                    Principal – Agency
                                    Problem

                                    The “Budget Constraint”
                                    of the Principal
          E



                                                F=Output
BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)


$
                                  Short – Run Profits Of
                                  The Principal




                                  Price Curve
                                  Short Run Average Cost
                                  Curve
                                  Marginal Cost Curve



                                              F=Output
    BILATERAL AGENCY - EXPLICIT
        Vertical Contract – (Principal – Agency)

• If the principal is operating in a
  perfectly competitive market outside of
  its relationship with the agent, its
  longrun profit function = 0
BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)


$
                                  Long Run Average Cost
                                  Curve




                                  Long Run Price Curve
                                  PROFITS = 0

                                  Marginal Cost Curve




                                             F=Output
       BILATERAL AGENCY - EXPLICIT
               Vertical Contract – (Principal – Agency)

• Long Run Profit Constraint

• F(a ) – W = 0
       2


  F(a ) – T – (1-a)F(a ) = 0
       2        2                2


 F – T – bF = 0
           2


 T = (1-b)F
   2
    BILATERAL AGENCY - EXPLICIT
        Vertical Contract – (Principal – Agency)

• The “complete” legal problem with the
  third constraint added is:

 L(a ) = F - T - bF
    1           2


       + l (bF – 1) +  (T + bF– a )
            2       2        2   2          2
 BILATERAL AGENCY - EXPLICIT
        Vertical Contract – (Principal – Agency)


T2



                                   A Contract Equilibrium
                                   Point


                                   Set of all feasible
                                   contracts




                                                b
    BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)

• As risk in output increases, either due to
  moral hazard or some third party cause,
  the risk reduces the marginal benefit of
  pay for performance β and thus causes
  the indifference curves to follow the
  feasible contract curve to the left.
 BILATERAL AGENCY - EXPLICIT
        Vertical Contract – (Principal – Agency)


T2



                                      A Contract Equilibrium
                                      Point


                                      Set of all feasible
                                      contracts




                                                   b
    BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)

• As productivity increases (technological
  innovation), the feasible contract curve
  stretches upward and the indifference
  curves follow the feasible contract curve to
  the right.
 BILATERAL AGENCY - EXPLICIT
        Vertical Contract – (Principal – Agency)


T2



                                   A Contract Equilibrium
                                   Point


                                   Set of all feasible
                                   contracts




                                                b
     BILATERAL AGENCY - EXPLICIT
       Vertical Contract – (Principal – Agency)

• As the Principal (firm) increases in size,
  either one of the two previous effects
  apply.
BILATERAL AGENCY - EXPLICIT
      Vertical Contract – (Principal – Agency)



• The “Principal-Agency ” exchange
  is the model featured in Cooter's
  treatment of contract law
BILATERAL AGENCY - EXPLICIT




            Double Moral
             Hazard
BILATERAL AGENCY - EXPLICIT
    Vertical Contract – (Principal – Agency)

Double Moral Hazard

Neither agent nor principal cannot
observe the effort or action applied by the
other.
This means the parties cannot know in
advance whether or not the contract will
be performed.
BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)




  In the double moral hazard version,
  both Principal and agent perform
  actions:
  L = aF - a1 - a2 + l1(aF1 - 1) + l2((1-a)F2 - 1)
  + 2(T2 + (1-a)F - a2)
BILATERAL AGENCY - EXPLICIT




            Competing
            Agents
  BILATERAL AGENCY - EXPLICIT
         Vertical Contract – (Principal – Agency)



U(F)
                                    A “perfectly competitive”
                                    risk neutral Principal
                                    contracts a “complete”
                                    contract with the agents

                                    In this case two
                                    “different” agents –
                                    “two” different contracts
          EH     EL



                                                F=Output

								
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