Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Exercise 11.3

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

March 2007 March 2007

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icroeconomics

Microeconom

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Ex 11.3(1): Question

purposepurpose: solution to an adverse selection problem: solution to an adverse selection problem methodmethod: find full-information solution from reservation utility levels. Then introduce : find full-information solution from reservation utility levels. Then introduce

incentive-compatibility constraint in order to find second-best solutionincentive-compatibility constraint in order to find second-best solution

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icroeconomics

Microeconom

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Ex 11.3(1): participation constraint

The principal knows the agent’s type The principal knows the agent’s type So maximises So maximises x x y y subject to subject to

where where = 0 = 0 for each individual typefor each individual type

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

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icroeconomics

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Ex 11.3(1): full-information case

Differentiate the binding participation constraint Differentiate the binding participation constraint to find the slope of the IC:to find the slope of the IC:

Since there is no distortion this slope must equal 1Since there is no distortion this slope must equal 1 This impliesThis implies

Using the fact that Using the fact that = = and substituting into the and substituting into the participation constraint:participation constraint:

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icroeconomics

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Ex 11.3(1): Full-information contracts

0

y

x

•

•

slope = 1

slope = 1

x*a = 2x*b = ½

y*b = ¼

y*a = 1

b_

_a

a-type’s reservation utilityb-type’s reservation utility

Space of (legal services, payment)

Contracts

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icroeconomics

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Ex 11.3(1): FI contracts, assessment

Solution has MRS = MRTSolution has MRS = MRT since there is no distortion… since there is no distortion… ……the allocation (the allocation (xx**aa, , yy**aa), (), (xx**bb, , yy**bb) is efficient) is efficient

We cannot perturb the allocation so as toWe cannot perturb the allocation so as to make one person better off…make one person better off… ……without making the other worse offwithout making the other worse off

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icroeconomics

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Ex 11.3 (2): Question

methodmethod: : Derive the incentive-compatibility constraintDerive the incentive-compatibility constraint Set up LagrangeanSet up Lagrangean Solve using standard methodsSolve using standard methods Compare with full-information values of Compare with full-information values of xx and and yy

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icroeconomics

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Ex 11.3 (2): “wrong” contract?

Now it is impossible to monitor the lawyer’s typeNow it is impossible to monitor the lawyer’s type Is it still viable to offer the efficient contracts Is it still viable to offer the efficient contracts ( (xx**aa, , yy**aa) ) andand

((xx**bb, , yy**bb))?? Consider situation of a type-Consider situation of a type-aa lawyer lawyer

if he accepts the contract meant for him he gets utilityif he accepts the contract meant for him he gets utility

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

So a type So a type aa would prefer to take… would prefer to take… a type-a type-bb contract contract rather than the efficient contractrather than the efficient contract

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Ex 11.3 (2): incentive compatibility

Given the uncertainty about lawyer’s type…Given the uncertainty about lawyer’s type… ……the firm wants to maximise expected profitsthe firm wants to maximise expected profits

it is risk-neutralit is risk-neutral

This must take account of the “wrong-contract” problem This must take account of the “wrong-contract” problem just mentionedjust mentioned

An An aa-type must be rewarded sufficiently… -type must be rewarded sufficiently… so that is not tempted to take a so that is not tempted to take a bb-type contract-type contract

The incentive-compatibility constraint for the The incentive-compatibility constraint for the aa types types

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Ex 11.3 (2): optimisation problem

Let Let be the probability that the lawyer is of type be the probability that the lawyer is of type aa Expected profits areExpected profits are

Structure of problem is as for previous exercises Structure of problem is as for previous exercises participation constraint for type participation constraint for type bb will be binding will be binding

incentive-compatibility constraint for type incentive-compatibility constraint for type aa will be binding will be binding

This enables us to write down the Lagrangean…This enables us to write down the Lagrangean…

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Ex 11.3 (2): Lagrangean

The Lagrangean for the firm’s optimisation problem is:The Lagrangean for the firm’s optimisation problem is:

where…where… is the Lagrange multiplier for is the Lagrange multiplier for bb’s’s participation constraint participation constraint is the Lagrange multiplier foris the Lagrange multiplier foraa’s’s incentive-compatibility incentive-compatibility

constraintconstraint

Find the optimum by examining the FOCs…Find the optimum by examining the FOCs…

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Ex 11.3 (2): Lagrange multipliers

Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to xxaa

and set result to 0and set result to 0

yields yields = = a a

Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to xxbb

and set result to 0 and set result to 0

using the value for using the value for this yields this yields = = b b

Use these values of the Lagrange multiplier in the Use these values of the Lagrange multiplier in the remaining FOCsremaining FOCs

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

Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to yyaa

and set result to 0and set result to 0

Substitute for Substitute for ::

Rearranging we findRearranging we find

exactly as for the full-information caseexactly as for the full-information case also MRS = 1, exactly as for the full-information casealso MRS = 1, exactly as for the full-information case illustrates the “no distortion at the top” principleillustrates the “no distortion at the top” principle

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icroeconomics

Microeconom

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

Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to yybb

and set result to 0and set result to 0

Substitute for Substitute for and and ::

Rearranging we findRearranging we find

this is less than this is less than ¼[¼[bb]]22… … ……the full-information income for a the full-information income for a bb-type-type

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Ex 11.3 (2): optimal x Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to

and set result to 0and set result to 0 get the get the b-b-type’s binding participation constrainttype’s binding participation constraint this yieldsthis yields

which becomeswhich becomes

Differentiate Lagrangean with respect to Differentiate Lagrangean with respect to and set result to 0 and set result to 0 get the get the a-a-type’s binding incentive-compatibility constrainttype’s binding incentive-compatibility constraint this yieldsthis yields

These are less than values for full-information contractsThese are less than values for full-information contracts for both for both aa-types and -types and bb-types-types

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Ex 11.3 (2): second-best solution

0xa

y

x

xb

yb

b_

_a

ya

•

• •

a-type’s reservation utilityb-type’s reservation utilitya-type’s full-info contractb-type’s second-best contracta-type’s second-best contract

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Ex 11.3: points to remember

Standard “adverse-selection” resultsStandard “adverse-selection” results Full-information solution is fully exploitativeFull-information solution is fully exploitative

binding participation constraint for both typesbinding participation constraint for both types Asymmetric informationAsymmetric information

incentive-compatibility problem for incentive-compatibility problem for aa-types-types Second best solutionSecond best solution

binding participation constraint for binding participation constraint for b-b-typetype binding incentive-compatibility constraint for binding incentive-compatibility constraint for a-a- type type no distortion at the topno distortion at the top