1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 11.1
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2007

2. Ex 11.1(1): Question
purpose: to illustrate and solve the “hidden information” problem
method: find full information solution, describe incentive-compatibility problem, then find second-best solution

3. Ex 11.1(1): Budget constraint
Consumer has income y and faces two possibilities
“not buy”: all y spent on other goods
“buy”: y  F(q) spent on other goods
Define a binary variable i:
i = 0 represents the case “not buy”
i = 1 represents the case “buy”
Then the budget constraint can be written
x + iF(q) ≤ y

4. Ex 11.1(2): Question 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

5. Ex 11.1(2): Preferences: qualitytb ta
(quality, other-goods) space
high-taste type
low-taste type
redraw in (quality, fee) space ta tb
preference
IC must be linear in t
ta > tb
Because linear ICs can only intersect once q
quality

6. Ex 11.1(2): Isoprofit curves, quality
(quality, fee) space
Iso-profit curve: low profits
lso-profit curve: medium profits
lso-profit curve: high profits
increasing
profit
P2 = F2  C(q)
Increasing, convex in quality
P1 = F1  C(q)
P0 = F0  C(q)

7. Ex 11.1(2): Full-information 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
Full-information solution, low type •
F*a
tbq
Type-a participation constraint taqa  Fa ≥0 •
F*b
Type-b participation constraint tbqb  Fb ≥0
Full information so firm can put each type on reservation IC q
q*b
q*a
quality

8. Ex 11.1(3,4): Question method: Set out nature of the problem
Describe in full the constraints
Show which constraints are redundant
Solve the second-best problem

9. Ex 11.1(3,4): Misrepresentation?F
Feasible set, high type
Feasible set, low type
Full-information solution
taq
Type-a consumer with a type-b deal •
F*a
preference
Type-a participation constraint taqa  Fa ≥0
tbq
Type-b participation constraint tbqb  Fb ≥0 •
F*b
A high type-consumer would strictly prefer the contract offered to a low type q
q*b
q*a
quality

10. Ex 11.1(3,4): background to problem
Utility obtained by each type in full-information solution is y
each person is on reservation utility level
given the U function, if you don’t consume the good you get exactly y
If a-type person could get a b-type contract
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
participation constraints
this incentive-compatibility problem

11. Ex 11.1(3,4): second-best problem
Participation constraint for the two types
taqa  Fa ≥ 0
tbqb  Fb ≥ 0
Incentive compatibility requires that, for the two types:
taqa  Fa ≥ taqb  Fb
tbqb  Fb ≥ tbqa  Fa
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
p[Fa  C(qa)] + [1  p][Fb  C(qb)] subject to
the participation constraints
the incentive-compatibility constraints
However, we can simplify the problem
which constraints are slack?
which are binding?

12. Ex 11.1(3,4): participation, b-types
First, we must have taqa  Fa ≥ tbqb  Fb
this is because
taqa  Fa ≥ taqb  Fb (a-type incentive compatibility) and
ta > tb (a-type has higher taste than b-type)
This implies the following:
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
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

13. Ex 11.1(3,4): participation, a-types
If Fb > 0 at the optimum, then qb > 0
follows from binding b-type participation constraint
tbqb  Fb = 0
This implies taqb  Fb > 0
because a-type has higher taste than b-type
ta > tb
This in turn implies taqa  Fa > 0
follows from a-type incentive-compatibility constraint
taqa  Fa ≥ taqb  Fb
So a-type participation constraint is slack and can be ignored

14. Ex 11.1(3,4): incentive compatibility, a-types
Could a-type incentive-compatibility constraint be slack?
could we have taqa  Fa > taqb  Fb ?
If so then it would be possible to increase Fa …
…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

15. Ex 11.1(3,4): incentive compatibility, b-types
Could b-type incentive-compatibility constraint be binding?
tbqa  Fa = tbqb  Fb ?
If so, then qa = qb
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”
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

16. Ex 11.1(3,4): Lagrangean Firm's problem is therefore
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

17. Ex 11.1(3,4): FOCs
Differentiate Lagrangean with respect to Fa and set result to zero:
p  m = 0
which implies m = p
Differentiate Lagrangean with respect to qa and set result to zero:
 pCq(qa) + mta = 0
given the value of m this implies Cq(qa) = ta
But this condition means, for the high-value a types:
marginal cost of quality = marginal value of quality
the “no-distortion-at-the-top” principle

18. Ex 11.1(3,4): FOCs (more)
Differentiate Lagrangean with respect to Fa and set result to zero:
1  p  l + m = 0
given the value of m this implies l = 1
Differentiate Lagrangean with respect to qb and set result to zero:
[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

19. Ex 11.1(3,4): Second-best solutionq F
quality
Feasible set for each type
Iso-profit contours
Contract for low type
taq
Contract for high type
preference •
Low type is on reservation IC, but MRS≠MRT Fa
tbq
High type is on IC above reservation level, but MRS=MRT • Fb qb
q*a

20. Ex 11.1: Points to remember
Full-information solution is bound to be exploitative
Be careful to specify which constraints are important in the second-best
Interpret the FOCs carefully