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Frank Cowell: Microeconomics
March 2007
Exercise 11.3
MICROECONOMICS
Principles and Analysis
Frank Cowell
Ex 11.3(1): Question
Frank Cowell: Microeconomics
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purpose: solution to an adverse selection problem
method: find full-information solution from reservation utility levels.
Then introduce incentive-compatibility constraint in order to find
second-best solution
Ex 11.3(1): participation constraint
Frank Cowell: Microeconomics
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The principal knows the agent’s type
So maximises x  y subject to
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where u = 0
for each individual type
In the full-information solution
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the participation constraint binds
there is no distortion
Ex 11.3(1): full-information case
Frank Cowell: Microeconomics
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Differentiate the binding participation constraint
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to find the slope of the IC:
Since there is no distortion this slope must equal 1
This implies
Using the fact that u = u and substituting into the
participation constraint:
Ex 11.3(1): Full-information contracts
Frank Cowell: Microeconomics
u
_b
y
Space of (legal services, payment)
a-type’s reservation utility
b-type’s reservation utility
Contracts
•
y*a = 1
slope = 1
u
_a
y*b = ¼
0
•
x*b
slope = 1
=½
x*a
x
=2
Ex 11.3(1): FI contracts, assessment
Frank Cowell: Microeconomics
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Solution has MRS = MRT
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since there is no distortion…
…the allocation (x*a, y*a), (x*b, y*b) is efficient
We cannot perturb the allocation so as to
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make one person better off…
…without making the other worse off
Ex 11.3 (2): Question
Frank Cowell: Microeconomics
method:
 Derive the incentive-compatibility constraint
 Set up Lagrangean
 Solve using standard methods
 Compare with full-information values of x and y
Ex 11.3 (2): “wrong” contract?
Frank Cowell: Microeconomics
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Now it is impossible to monitor the lawyer’s type
Is it still viable to offer the efficient contracts (x*a, y*a) and
(x*b, y*b)?
Consider situation of a type-a lawyer
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if he accepts the contract meant for him he gets utility
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but if he were to get a type-b contract he would get utility
So a type a would prefer to take…
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a type-b contract
rather than the efficient contract
Ex 11.3 (2): incentive compatibility
Frank Cowell: Microeconomics
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Given the uncertainty about lawyer’s type…
…the firm wants to maximise expected profits
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This must take account of the “wrong-contract” problem
just mentioned
An a-type must be rewarded sufficiently…
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it is risk-neutral
so that is not tempted to take a b-type contract
The incentive-compatibility constraint for the a types
Ex 11.3 (2): optimisation problem
Frank Cowell: Microeconomics
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Let p be the probability that the lawyer is of type a
Expected profits are
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Structure of problem is as for previous exercises
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participation constraint for type b will be binding
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incentive-compatibility constraint for type a will be binding
This enables us to write down the Lagrangean…
Ex 11.3 (2): Lagrangean
Frank Cowell: Microeconomics
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The Lagrangean for the firm’s optimisation problem is:
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where…
l is the Lagrange multiplier for b’s participation constraint
m is the Lagrange multiplier fora’s incentive-compatibility
constraint
Find the optimum by examining the FOCs…
Ex 11.3 (2): Lagrange multipliers
Frank Cowell: Microeconomics
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Differentiate Lagrangean with respect to xa
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and set result to 0
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yields m = pta
Differentiate Lagrangean with respect to xb
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and set result to 0
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using the value for m this yields l = tb
Use these values of the Lagrange multiplier in the
remaining FOCs
Ex 11.3 (2): optimal payment, a-types
Frank Cowell: Microeconomics
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Differentiate Lagrangean with respect to ya
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and set result to 0
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Substitute for m:
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Rearranging we find
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exactly as for the full-information case
also MRS = 1, exactly as for the full-information case
illustrates the “no distortion at the top” principle
Ex 11.3 (2): optimal payment, b-types
Frank Cowell: Microeconomics
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Differentiate Lagrangean with respect to yb
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and set result to 0
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Substitute for l and m:
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Rearranging we find
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this is less than ¼[tb]2…
…the full-information income for a b-type
Ex 11.3 (2): optimal x
Frank Cowell: Microeconomics
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Differentiate Lagrangean with respect to l
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and set result to 0
get the b-type’s binding participation constraint
this yields
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which becomes
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Differentiate Lagrangean with respect to m
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and set result to 0
get the a-type’s binding incentive-compatibility constraint
this yields
These are less than values for full-information contracts
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for both a-types and b-types
Ex 11.3 (2): second-best solution
Frank Cowell: Microeconomics
a-type’s reservation utility
b-type’s reservation utility
a-type’s full-info contract
b-type’s second-best contract
a-type’s second-best contract
u
_b
y
•
^ya
•
u
_a
^yb
0
•
x
^xb
^xa
Ex 11.3: points to remember
Frank Cowell: Microeconomics
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Standard “adverse-selection” results
Full-information solution is fully exploitative
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Asymmetric information
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binding participation constraint for both types
incentive-compatibility problem for a-types
Second best solution
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binding participation constraint for b-type
binding incentive-compatibility constraint for a- type
no distortion at the top