Transcript Extensive Form - London School of Economics

Frank Cowell: Microeconomics
March 2007
Exercise 11.1
MICROECONOMICS
Principles and Analysis
Frank Cowell
Ex 11.1(1): Question
Frank Cowell: Microeconomics


purpose: to illustrate and solve the “hidden information” problem
method: find full information solution, describe incentive-compatibility
problem, then find second-best solution
Ex 11.1(1): Budget constraint
Frank Cowell: Microeconomics

Consumer has income y and faces two possibilities
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Define a binary variable i:
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“not buy”: all y spent on other goods
“buy”: y  F(q) spent on other goods
i = 0 represents the case “not buy”
i = 1 represents the case “buy”
Then the budget constraint can be written

x + iF(q) ≤ y
Ex 11.1(2): Question
Frank Cowell: Microeconomics
method:
 First draw ICs in space of quality and other goods
 Then redraw in space of quality and fee
 Introduce iso-profit curves
 Full-information solutions from tangencies
Ex 11.1(2): Preferences: quality
Frank Cowell: Microeconomics
(quality, other-goods) space
F
x
high-taste type
low-taste type
redraw in (quality, fee) space
ta
tb
ta
tb
IC must be linear in t
ta > tb
Because linear ICs can
only intersect once
q
quality
Ex 11.1(2): Isoprofit curves, quality
Frank Cowell: Microeconomics
(quality, fee) space
F
Iso-profit curve: low profits
lso-profit curve: medium profits
lso-profit curve: high profits
P2 = F2  C(q)
Increasing, convex in
quality
P1 = F1  C(q)
P0 = F0  C(q)
q
quality
Ex 11.1(2): Full-information
Frank Cowell: Microeconomics
solution
reservation IC, high type
F
Firm’s feasible set for a high type
Reservation IC + feasible set, low
type
lso-profit curves
taq Full-information solution, high type
•
F*a
F*b
Full-information solution, low type
tbq
Type-a participation
constraint taqa  Fa ≥0
•
Type-b participation
constraint tbqb  Fb ≥0
q
q*b
q*a
quality
Full information so
firm can put each type
on reservation IC
Ex 11.1(3,4): Question
Frank Cowell: Microeconomics
method:
 Set out nature of the problem
 Describe in full the constraints
 Show which constraints are redundant
 Solve the second-best problem
Ex 11.1(3,4): Misrepresentation?
Frank Cowell: Microeconomics
Feasible set, high type
F
Feasible set, low type
Full-information solution
taq Type-a consumer with a type-b deal
•
F*a
F*b
Type-a participation
constraint taqa  Fa ≥0
tbq
Type-b participation
constraint tbqb  Fb ≥0
•
q
q*b
q*a
quality
A high type-consumer
would strictly prefer the
contract offered to a low
type
Ex 11.1(3,4): background to problem
Frank Cowell: Microeconomics

Utility obtained by each type in full-information solution is y
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If a-type person could get a b-type contract
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each person is on reservation utility level
given the U function, if you don’t consume the good you get exactly y
a-type’s utility would then be
taq*b  F*b +y
given that tbq*b  F*b = 0…
…a-type’s utility would be [ta  tb]q*b + y >y
So an a-type person would want to take a b-type contract
In deriving second-best contracts take account of
1.
2.
participation constraints
this incentive-compatibility problem
Ex 11.1(3,4): second-best problem
Frank Cowell: Microeconomics

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Participation constraint for the two types
 taqa  Fa ≥ 0
b b
b
 t q  F ≥ 0
Incentive compatibility requires that, for the two types:
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Suppose there is a proportion p, 1 p of a-types and b-types
Firm's problem is to choose qa, qb, Fa and Fb to max expected profits

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taqa  Fa ≥ taqb  Fb
tbqb  Fb ≥ tbqa  Fa
p[Fa  C(qa)] + [1  p][Fb  C(qb)] subject to
the participation constraints
the incentive-compatibility constraints
However, we can simplify the problem
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which constraints are slack?
which are binding?
Ex 11.1(3,4): participation, b-types
Frank Cowell: Microeconomics

First, we must have taqa  Fa ≥ tbqb  Fb

1.
2.

This implies the following:
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if tbqb  Fb > 0 (b-type participation slack)
then also taqa  Fa > 0 (a-type participation slack)
But these two things cannot be true at the optimum
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
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this is because
taqa  Fa ≥ taqb  Fb (a-type incentive compatibility) and
ta > tb (a-type has higher taste than b-type)
if so it would be possible for firm to increase both Fa and Fb
thus could increase profits
So b-type participation constraint must be binding

tbqb  Fb = 0
Ex 11.1(3,4): participation, a-types
Frank Cowell: Microeconomics

If Fb > 0 at the optimum, then qb > 0

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This implies taqb  Fb > 0

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because a-type has higher taste than b-type
ta > tb
This in turn implies taqa  Fa > 0



follows from binding b-type participation constraint
tbqb  Fb = 0
follows from a-type incentive-compatibility constraint
taqa  Fa ≥ taqb  Fb
So a-type participation constraint is slack and can be
ignored
Frank Cowell: Microeconomics
Ex 11.1(3,4): incentive
compatibility, a-types
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Could a-type incentive-compatibility constraint be slack?
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If so then it would be possible to increase Fa …
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could we have taqa  Fa > taqb  Fb ?
…without violating the constraint
this follows because a-type participation constraint is slack
taqa  Fa > 0
So a-type incentive-compatibility must be binding
 taqa  Fa = taqb  Fb
Frank Cowell: Microeconomics
Ex 11.1(3,4): incentive
compatibility, b-types
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Could b-type incentive-compatibility constraint be binding?
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If so, then qa = qb
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follows from fact that a-type incentive-compatibility constraint is binding
taqa  Fa = taqb  Fb
which, with the above, would imply [tb  ta]qa = [tb  ta]qb
given that ta > tb this can only be true if qa = qb
So, both incentive-compatibility conditions bind only with “pooling”
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

tbqa  Fa = tbqb  Fb ?
but firm can do better than pooling solution:
increase profits by forcing high types to reveal themselves
So the b-type incentive-compatibility constraint must be slack


tbqb  Fb > taqb  Fb
…and it can be ignored
Ex 11.1(3,4): Lagrangean
Frank Cowell: Microeconomics
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Firm's problem is therefore


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max expected profits subject to..
…binding participation constraint of b type
…binding incentive-compatibility constraint of a type
Formally, choose qa, qb, Fa and Fb to max

p[Fa  C(qa)] + [1  p][Fb  C(qb)]
 + l[tbqb  Fb]
 + m[taqa  Fa  taqb +Fb]

Lagrange multipliers are


l for the b-type participation constraint
m for the a-type incentive compatibility constraint
Ex 11.1(3,4): FOCs
Frank Cowell: Microeconomics

Differentiate Lagrangean with respect to Fa and set result to
zero:
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Differentiate Lagrangean with respect to qa and set result to
zero:
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pm=0
which implies m = p
 pCq(qa) + mta = 0
given the value of m this implies Cq(qa) = ta
But this condition means, for the high-value a types:
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marginal cost of quality = marginal value of quality
the “no-distortion-at-the-top” principle
Ex 11.1(3,4): FOCs (more)
Frank Cowell: Microeconomics

Differentiate Lagrangean with respect to Fa and set result to
zero:
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Differentiate Lagrangean with respect to qb and set result to
zero:
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1pl+m=0
given the value of m this implies l = 1
[1 p]Cq(qb) + ltb  mta = 0
given the values of l and m this implies Cq(qa) = ta
[1 p]Cq(qb) + tb  pta = 0
Rearranging we find for the low-value b-types

marginal cost of quality < marginal value of quality
Ex 11.1(3,4): Second-best solution
Frank Cowell: Microeconomics
Feasible set for each type
F
Iso-profit contours
Contract for low type
taq Contract for high type
•
Fa
Low type is on
reservation IC, but
MRS≠MRT
tbq
High type is on IC
above reservation level,
but MRS=MRT
Fb
•
q
qb
q*a
quality
Ex 11.1: Points to remember
Frank Cowell: Microeconomics
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Full-information solution is bound to be
exploitative
Be careful to specify which constraints are
important in the second-best
interpret the FOCs carefully