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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

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
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
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
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
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

• 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:
• Landlord – Tenant leases
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
• 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
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
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

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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