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Exercise 11.3

MICROECONOMICS

Principles and Analysis

Frank Cowell

March 2007

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Ex 11.3(1): Question

purpose: solution to an adverse selection problem

method: find full-information solution from reservation utility levels. Thenintroduce incentive-compatibility constraint in order to find second-best solution

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Ex 11.3(1): participation constraint

The principal knows the agents type

So maximisesxysubject to

where u = 0 for each individual type

In the full-information solution the participation constraint binds there is no distortion

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Ex 11.3(1): full-information case

Differentiate the binding participation constraint

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:

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Ex 11.3(1): Full-information contracts

0

y

x

slope = 1

slope = 1

x*a = 2x*b =

y*b =

y*a = 1

ub_

_ua

a-types reservation utility

b-types reservation utility

Space of (legal services, payment)

Contracts

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Ex 11.3(1): FI contracts, assessment

Solution has MRS = MRT

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 make one person better off

without making the other worse off

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Ex 11.3 (2): Question

method:

Derive the incentive-compatibility constraint Set up Lagrangean

Solve using standard methods

Compare with full-information values ofxand y

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Ex 11.3 (2): wrong contract?

Now it is impossible to monitor the lawyers 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 if he accepts the contract meant for him he gets utility

but if he were to get a type-b contract he would get utility

So a type awould prefer to take

a type-b contract

rather than the efficient contract

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Ex 11.3 (2): incentive compatibility

Given the uncertainty about lawyers type

the firm wants to maximise expected profits

it is risk-neutral

This must take account of the wrong-contract problem

just mentioned

An a-type must be rewarded sufficiently

so that is not tempted to take a b-type contract

The incentive-compatibility constraint for the a types

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Ex 11.3 (2): optimisation problem

Let p be the probability that the lawyer is of type a

Expected profits are

Structure of problem is as for previous exercises

participation constraint for type b will be binding

incentive-compatibility constraint for type a will be binding

This enables us to write down the Lagrangean

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Ex 11.3 (2): Lagrangean

The Lagrangean for the firms optimisation problem is:

where

l is the Lagrange multiplier for bsparticipation constraint

m is the Lagrange multiplier forasincentive-compatibility

constraint

Find the optimum by examining the FOCs

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Ex 11.3 (2): Lagrange multipliers

Differentiate Lagrangean with respect toxa

and set result to 0

yields m = pta

Differentiate Lagrangean with respect toxb

and set result to 0

using the value for m this yields l = tb

Use these values of the Lagrange multiplier in theremaining FOCs

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Ex 11.3 (2): optimal payment, a-types

Differentiate Lagrangean with respect to ya

and set result to 0

Substitute for m:

Rearranging we find

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

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Ex 11.3 (2): optimal payment, b-types

Differentiate Lagrangean with respect to yb

and set result to 0

Substitute for l and m:

Rearranging we find

this is less than [tb]2

the full-information income for a b-type

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Ex 11.3 (2): optimalx

Differentiate Lagrangean with respect to l

and set result to 0

get the b-types binding participation constraint

this yields

which becomes

Differentiate Lagrangean with respect to m

and set result to 0

get the a-types binding incentive-compatibility constraint

this yields

These are less than values for full-information contracts for both a-types and b-types

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Ex 11.3 (2): second-best solution

0 ^xa

y

x^xb

^yb

ub_

_ua

^ya

a-types reservation utilityb-types reservation utility

a-types full-info contract

b-types second-best contract

a-types second-best contract

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Ex 11.3: points to remember

Standard adverse-selection results

Full-information solution is fully exploitative

binding participation constraint for both types

Asymmetric information incentive-compatibility problem for a-types

Second best solution

binding participation constraint for b-type

binding incentive-compatibility constraint for a- type no distortion at the top