Laffont Martimort The Theory of Incentives, Summary, Preliminary Draft. Moral Hazard Notes Summary of Theory of Incentives by Laffont and Martimort 2014. Exogeneous Endogenous First Best Second Best Contracts Asymmetric Information #economics #microeconomics #microeconomic analysis #economicpolicy #contracts #incentives #students #university #lagrangian #Lagrange #multipliers #optimization #optimum #pareto adverse selection wiki adverse selection pdf adverse selection and moral hazard adverse selection definition adverse selection definizione selezione avversa economia market of lemons azzardo morale

1. Notes on
Adverse Selection
Preliminary Draft
Summary based on The Theory of Incentives, Laffont and Martimort (2014)
Typos and small mistakes may be present, and they are entirely mine.
1

2. Overview
 Basic Model
 Complete Information Optimal Contract, FB and implementation
 Incentive feasible menu of Contracts
 Special cases: pooling and shutdown
 Information Rents
 Optimization program of the principal
 Rent Extraction efficiency trade off: optimal contract under a.s.
 SB graphical interpretation
 Shutdown Policy
 More general agent’s utility function, optimal contract
 Contract Theory at work: Nonlinear monopoly pricing and financial contracts
2

3. Basics of the Model
 Consider a Principal who delegates the production of q units of the good to an
agent.
 Utility of the Principal is S(q) with positive and diminishing marginal revenue:
S’(q)>0 and S’’(q)<0.
 Production cost is not observable to the principal = hidden knowledge of the
agent. The marginal cost can be either high or low, depending if the agent is
efficient (𝜃−) or inefficient(𝜃−
). F is fixed cost.
 𝐶 𝑞, 𝜃− = 𝜃− 𝑞 + 𝐹 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑣
 𝐶 𝑞, 𝜃−
= 𝜃−
𝑞 + 𝐹 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 1 − 𝑣
 Spread of uncertainty on agent’s marginal cost: Δ𝜃 = 𝜃−
− 𝜃−
 Under adverse selection: uncertainty on agent’s type is exogenous.
3

4.  Quantity produced and the transfer t received by the Agent from the
principal are the economic variables of the problem.
 A is the set of feasible allocations.
 A=(q,t): q 𝜖 𝑅+, 𝑡 𝜖 𝑅) then q is nonnegative, t can be negative.
 Variables are observable by a court of law.
Timing of the problem:
4
A discovers
his type
P offers a
contract
A accepts of
rejects the
contract
The
contract is
executed
0
1
2
3

5. First Best Production Levels
(Complete Information)
 Suppose Complete information. Efficient production is obtained equating
MR and MC.
 𝑆′ 𝑞− ∗ = 𝜃−
 𝑆′
𝑞− ∗ = 𝜃−
 These production levels are carried out if their social values are non
negative:
 𝑊−∗
= 𝑆(𝑞∗
) − 𝜃−
𝑞−∗
− 𝐹 ≥ 0 and 𝑊−
∗
= 𝑆 𝑞−
∗
− 𝜃− 𝑞− − 𝐹 ≥ 0
 Social value of the efficient is greater than inefficient. 𝑊_>𝑊−
 𝑆 𝑞−
∗ − 𝜃− 𝑞− ≥ 𝑆′ 𝑞−∗ − 𝜃− 𝑞−∗ ≥ 𝑆 𝑞−∗ − 𝜃− 𝑞−∗
 W=MR-MC of «efficient agent» is greater than «efficient mimicking
inefficient», which in turn is greater than «inefficient».
5

6. FB implementation:
PC in complete information
 Principal must offer the agent a utility level that is at least as high as he
would obtain outside the relationship. This is called Participation
Constraint.
 𝑡− − 𝜃− 𝑞− ≥ 0
 𝑡−
− 𝜃−
𝑞−
≥ 0
 Principal offers a «take it or leave it» offer to the agent:
 If agent is inefficient the principal offers a transfer 𝑡− = 𝜃− 𝑞−∗
 If agent is efficient the principal offers a transfer 𝑡− = 𝜃− 𝑞−
∗
 Whatever its type agent accepts and makes zero profit;
 Under complete information delegation is costless for the principal.
6

7. Indifference curves of both types
0
2
4
6
8
10
Inefficient agents have steeper slopes, since theta is higher
u inefficient u efficient u inefficient2 u efficient2 u inefficient3 u inefficient4
7
Spence Mirrlees Property: indifference curves have single crossing.
Utilities increases for the agent in northwest direction; utility of the principal increases
in south.east direction

8. First Best Graph
8
A* and B* are the optimal contract
where strictly concave indifference
curve of the principal is tangent to the
zero rent isoutility curve of the
corresponding type.
Isoutility of the principal increases in
the southeast direction, of the agent in
northwest direction.
Principal has all bargaining power
under complete information so V=W.

9. Incentive Feasible
Menu of Contracts
 Both types prefer B* to A*.
 Efficient isoutility curve passing
through B* gives a positive utility to
him, while A* gives zero utility. At
same time, to the Inefficient B* gives
zero utility and A* gives negative
utility.
 The efficient type prefers B and
mimicks the inefficient to profit from
this information rent.
 The menu of contracts of complete
information is not incentive
compatible for adverse selection.
 Now the efficient produces les:s even
if obtains a lower transfer from the
principal, efficient agent gains from
gap of costs and mimicks the other.
9

10. SB implementation: IC
 A menu of contracts is incentive feasible if the contract with high transfers
and higher quantity is weakly preferred by efficient and the contract with
low transfer and lower quantity is weakly preferred by inefficient.
Incentive Compatibility Constraint
 𝑡− − 𝜃− 𝑞− ≥ 𝑡− − 𝜃− 𝑞− (the binding contraint IC of the efficient)
 𝑡−
− 𝜃−
𝑞−
≥ 𝑡− − 𝜃−
𝑞−
Participation Constraint
 𝑡− − 𝜃− 𝑞− ≥ 0
 𝑡− − 𝜃− 𝑞− ≥ 0 (the binding contraint PC of inefficient)
10

11. Special Cases: Pooling and Shutdown
 Pooling of contracts: when both contracts for each type coincide: transfers given and
quantity produced are same for both types.
 IC satisfied, but there is loss of flexibility in allocations. Also PC is satisfied since the
inefficient entering into contract implies that also efficient enters.
 Shutdown of the Inefficient: The inefficient has the null contract (0,0)
 And the non zero contract (t,q) is carried out by efficient agent.
 Efficient IC and PC reduces to a PC 𝑡 − 𝜃𝑞 ≥ 0
 IC of inefficient is 0 ≤ 𝑡 − 𝜃−
𝑞
11
Pro: reduces number of constraints. Cons: principal gives up production if agent is
inefficient. Excessive Screening costs of subset of types.

12. Monotonicity Constraints
 Quantity of the efficient agent is higher than inefficient agent both IN SB
and FB.
 Implementability condition 𝒒− ≥ 𝒒−
 Any pair of outputs which are implementable (quantities associated to IC
contract) must satisfy this condition.
 Transfers associated to IC :
𝜃−
𝑞−
− 𝑞− ≤ 𝑡−
− 𝑡− ≤ 𝜃−(𝑞−
− 𝑞−)
12

13. Information Rents
 Under complete information principal decrease utility of agents to zero.
U_=0 and u^-=0
 No longer done under adverse selection: efficient mimicks the inefficient
agent and the principal is no longer able to verify which type actually is.
Agent profits from the gap in costs: obtains the same high transfer of the
inefficient, even if he had a lower cost of production.
 As a solution, the principal gives up some information rent to the agent,
but he is still able to reduce information rent up to a point.
 Utility of the efficient mimicking the inefficient is : equal to the inefficient +
information rent.
𝒕−
− 𝜽− 𝒒−
≥ 𝒕−
− 𝜽−
𝒒−
+ 𝚫𝜽𝒒−
= 𝑼−
+ 𝚫𝜽𝒒−
13

14. SB Optimal Contract
 Max 𝑣(𝑆 𝑞− − 𝑡−) + 1 − 𝑣 𝑆(𝑞− − 𝑡−) where v is probability of being efficient.
 New optimization variables are now U and q. and U=t-thetaq.
 Since U=t-thetaq, the transfer is equal to thetaq-u: we replace it in the objective fx.
 max 𝑣 𝑆 𝑞− − 𝜃− 𝑞− + 1 − 𝑣 𝑆 𝑞−
− 𝜃−
𝑞−
− (𝒗𝑼− + 𝟏 − 𝒗 𝑼−
).
 IC and PC: Utility of the efficient is greater than inefficient and info rent:
 U_>𝑼− + 𝚫𝜽𝒒− ;
 𝑼→ ≥ 𝑼− − 𝚫𝜽𝒒−;
 U_>0
 𝑼−>0
14

15. Constraints Relaxation
 Information rents let us analyse the distributive impact of asymmetric
information, allocative efficiency and gains from trade.
 Lagrangian can be used but we need to check that the problem is concave. We
can also check ex post that concavity is satisfied.
 If the inefficient mimicks the efficient the PC of efficient is always satisfied. What
we have to investigate is the INEFFICIENT PC.
 Moreover, since B is preferred by both types, there is no possibility that the
inefficient mimicks the efficient, since he has higher costs.
 So IC contraint binding is EFFICIENT IC.
15

16. SB SOLUTIONS
 Utility is now information rent for efficient agent and zero for inefficient.𝑈− =
Δ𝜃𝑞− 𝑎𝑛𝑑 𝑈− = 0
 New maximization of principal: unique information rent new constraint.
 𝑀𝐴𝑋 𝑣(𝑆 𝑞 − 𝜃𝑞− + (1 − 𝑣)(𝑆 𝑞− − 𝜃− 𝑞− − 𝑣Δ𝜃𝑞−
 Sb Solution: MR = MC : 𝑆′ 𝑞− = 𝜃 and quantity sb = quantity fb.
 Increasing Inefficient output increases efficiency but also information rent and principal
payoff is reduced by 𝑣Δ𝜃𝑑𝑞:
 1 − 𝑣 𝑆′ 𝑞− − 𝜃− = 𝑣Δ𝜃; at SB principal does not increase or decrease the inefficient
quantity. This is Rent Extraction / Efficiency Trade-Off.
 𝑞−
𝑆𝐵 = 𝑞−
∗ > 𝑞− ∗> 𝑞− 𝑆𝐵
16

17. SB SOLUTIONS
 Under Asymmetric information optimal menu of contracts entails:
 No output distortion for efficient agent wrt FB and
 Downward distortion for the inefficient type with a lower q wrt to FB.
𝑆′ 𝑞− 𝑠𝑏 = 𝜃− +
𝑣
1 − 𝑣
∗ Δ𝜃
 Only efficient obtains positive information rent 𝑈− = Δ𝑞−𝑠𝑏
SB TRANSFERS ARE GIVEN BY
 𝑡=𝜃− 𝑞−
∗ +Δ𝜃𝑞−𝑆𝐵 :efficient gets compensation from FB and info rent.
 𝑡−𝑠𝑏
= 𝜃−
𝑞− 𝑠𝑏
: inefficient gets compensation from SB cost (lower).
17

18. SB Graph 18
 FB solution (AB) was not IC.
 SB (B*,C) IC: same production levels
and higher transfer for efficient
agent.
 Efficient isoutility passes through C,
thus Eff. Is indifferent between B*
and C but C gives higher transfer.
 Rent given up to efficient is delta
theta q upper bar.
 Inefficient agent’s output is reduced
such that Principal gain is higher
than efficiency cost.

19. Final Optimal SB
Menu of Contract:
(Asb, Bsb)
19
 The optimal trade off solution.
 A fb = C quantity. Efficient q is
same.
 A sb gives higher transfer than
B fb but lower than C.
 B sb gives lower transfer than B*
but a lower quantity.

20. Shutdown Policy
 If First order solution has no positive solution for inefficient, quantity of SB of inefficient
should be set at zero.
 B SB =0; A SB= A FB
 𝑣Δ𝜃𝑞−𝑆𝐵 ≥ 1 − 𝑉 𝑆 𝑞−𝑆𝐵 − 𝜃− 𝑞−𝑠𝑏)
Shutdown policy is optimal when the expeced utiliy of the principal from efficient FB is
greater than efficient and inefficient SB.
 𝑣(𝑆 𝑞∗ − 𝜃− 𝑞−
∗
≥ 𝑣 𝑎𝑛𝑑 (1 − 𝑣)( 𝑆 𝑞 𝑠𝑏 − 𝜃− 𝑞−
𝑠𝑏
− Δ𝜃𝑞−𝑠𝑏
+ (1 − 𝑣)(𝑆 𝑞−𝑠𝑏
− 𝜃−
𝑞−𝑠𝑏
)
 𝑞−
∗ = 𝑞−
𝑠𝑏 when 𝑣𝛥𝜃𝑞−𝑆𝐵 ≥ 1 − 𝑉 𝑆 𝑞−𝑆𝐵 − 𝜃− 𝑞−𝑠𝑏)
20

21. 2,10 More General Utility function
 Instead of writing 𝜃𝑞 we write a function C(q,theta).
 Same IC and PC, agent’s utility: 𝑈 = 𝑡 − 𝐶(𝑞, 𝜃)
 𝜙 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑖𝑛 𝑐𝑜𝑠𝑡𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 = 𝐶 𝑞−, 𝜃− − 𝐶(𝑞,𝜃
−
)
 Optimal Contract is then
 𝑈− ≥ 𝑈−
+ 𝜙 𝑞−
 𝑈− = Φ 𝑞− since MC of phi is positive reducing inefficient agent’s output
reduces efficient agent’s information rent.
 No output distortion for efficient MR= MC 𝑆′
𝑞_ = 𝐶′
𝑞−, 𝜃−
 Downard distortion for inefficient
21

22. Optimal Contracts
 𝐼𝑓 𝐶 𝑞 𝜃
> 0 𝑠𝑒𝑒 𝑛𝑒𝑥𝑡 𝑠𝑙𝑖𝑑𝑒 the optimal contract entails:
 No output distortion with respect to the first best outcome for the efficient type
 𝑞−
𝑠𝑏
= 𝑞−
∗
with 𝑆′
𝑞−
∗
= 𝐶 𝑞(𝑞−, 𝜃−)
 A downward distortion for the less efficient type:
 FB was 𝑆′
𝑞∗
= 𝐶 𝑞 𝑞−∗
𝑆𝐵 ∶ 𝑆′
𝑞−𝑠𝑏
= 𝐶 𝑄 𝑞−
, 𝜃−
+
𝑣
1−𝑣
𝜙′
𝑞−𝑠𝑏
 Only efficient type gets a positive information rent given by
𝑈−𝑠𝑏 = 𝜙(𝑞− 𝑠𝑏)
Second Best Transfers are:
𝑡− = 𝐶 𝑞∗, 𝜃− + 𝜙 𝑞−𝑠𝑏 𝑎𝑛𝑑 𝑡−𝑠𝑏 = 𝐶(𝑞−𝑠𝑏, 𝜃−)
The transfer with t efficient has cost associated to the first best solution and phi associated
to inefficient quantity; the transfer of the inefficient is just q inefficient.
22

23. Spence Mirrlees Property
 It indicates a single crossing of isoutility curves;
 It is a constant sign condition on the derivative of C wrt to q and theta. Moreover it implies
that phi has positive first derivative.
 In fact : 𝐶 𝑞 > 0, 𝐶 𝜃 > 0, 𝐶 𝑞𝑞 > 0, 𝐶 𝑞𝑞𝜃 > 0. SM generalizes to 𝐶 𝑞𝜃 >0.
 If the Marginal Cost <0 the output distortion of the inefficient was upwards. In such a model
the FB would be lower than SB and information rent would be the same, but an increase in
quantity of inefficient would increase the rent.
 In our solution instead we have a positive marginal cost with a downward distortion of the
inefficient and Sb quantity of inefficient is lower than FB; information rent with a decrease in
quantity of inefficient decreasing the information rent of the efficient.
 It guarantees that only efficient type IC has to be taken into account.
23
𝜙′
𝑞 > 0 𝑎𝑛𝑑 𝐶 𝑞 𝜃
> 0; 𝜙 𝑞 𝑠𝑏
≥ 𝜙(𝑞−
𝑠𝑏
)

24. Non Linear Pricing by Monopoly
 Maskin and Riley (1984): principal is seller of a good with production cost cq. Consumers are
continuum of buyers thus can be seen as a single agent.
 Principal’s Utility: V=t-cq
 Consumers utility are 𝑈 = 𝜃𝑢 𝑞 − 𝑡; suppose that parameter theta is drawn independently
from the same distribution and v is probability that type theta upper bar is drawn from the
distribution by the Law of Large Numbers.
 Information Rents are 𝑈− = 𝑢 𝑄− − 𝑡− ; 𝑈− = 𝑢(𝑞− − 𝑡−)
 Low valuation agent is «the efficient» denoted with the lower bar, the highest valuation agent
is «the inefficient» with the upper bar on his variables.
24

25. Non Linear Monopoly Pricing
 max 𝑣 𝜃− − 𝑢 𝑞 − 𝑐𝑞− + 1 − 𝑣 𝜃− 𝑢 𝑞− − 𝑐𝑞− − (𝑣𝑈− − 1 − 𝑣 𝑈−)
 Subject to:
 𝑈− ≥ 𝑈− − Δ𝜃𝑢 𝑞− ; 𝑈− ≥ 𝑈− + Δ𝜃 𝑞− binding ; 𝑈− ≥ 0 binding and 𝑈− ≥ 0
 Efficient type is the one with highest valuation for the good theta upper bar. Hence
𝑈− ≥ 𝑈− + Δ𝜃 𝑞− and 𝑈− ≥ 0 are binding.
 No output distortion for high valuation type 𝑞−𝑠𝑏
= 𝑞−∗
and 𝜃−
𝑞−∗
= 𝑐
 Downward distortion for low valuation agent’s output wrt to FB;
 𝑞−
𝑠𝑏
> 𝑞−
∗ and 𝜃−
𝑞−∗
= 𝑐 and 𝑐 𝑠𝑏
= (𝜃−−
𝑣
1−𝑣
Δ𝜃)(𝑢′
𝑞−
𝑠𝑏
 So unit pricec is different if buyers demand q^-FB or q_SB: price are nonlinear.
25

26. Quality and Price Discrimination - i
 Mussa and Rosen (1978): agents buy one unit of a commodity with quality q but are vertically
differentiated with respect to their preferences for the good the original cost of producing one unit of
quality q is C(q); the prinicpal has the utility function V=t-c(Q)
 Agent’s Utility: 𝑢 = 𝜃𝑞 − 𝑡
 IC and PC are rewritten in form of information rents. Instead of u(q) it deals with q directly.
 𝑈− ≥ 𝑈− − Δ𝜃 𝑞−
 𝑈−
≥ 𝑈− + Δ𝜃 𝑞− binding
 𝑈− ≥ 0 binding
 𝑈− ≥ 0
 max 𝑣 𝜃 − 𝑢 𝑞 − 𝑐𝑞 + 1 − 𝑣 𝜃− 𝑢 𝑞− − 𝑐𝑞− − (𝑣𝑈− + 1 − 𝑣 𝑈−)
 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑣 𝑜𝑓 𝑏𝑒𝑖𝑛𝑔 𝑖𝑛𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡.
26

27. Quality and Price Discrimination - ii
 Highest valuation agent receives the FB quality q^sb=q^fb and theta=mc(q^− fb)
 Lowest valuation agent receives the quality below FB. q_sb<q_fb.
 𝜃− = 𝐶′ 𝑞−
𝑠𝑏 +
𝑣
1−𝑣
Δ𝜃) 𝑎𝑛𝑑 𝜃− = 𝐶′(𝑞−
∗ ) ; spectrum of qualities is larger under
asymmetric information than under complete information.
 Incentive of the seller to put on the market low quality good is well documened
and damaging its own goods may be part of firm’s optimal selling strategy when
screening consumers’ willingness to pay for quality.
27

28. Financial Contracts
 Freixas and Laffont (1990) principal is a lender who provides a loan of size k to a borrower.
 Capital cost: Rk to the lender, since it could be invested at risk free rate R.
 Lender’s utility: V= t-Rk
 Borrower makes a profit 𝑈 = 𝜃𝑓 𝑘 − 𝑡 where t is the repayment to the lender and f(k) is the
production with k units of capital.
 Theta is a parameter on productivity shock. 𝜃−
ℎ𝑎𝑠 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑣 𝑎𝑛𝑑 𝜃− 1 − 𝑣 𝑡𝑜 𝑜𝑐𝑐𝑢𝑟.
 IC and PC are written as borrower’s information rents 𝑈− = 𝜃− 𝑓 𝑘− − 𝑡−
 𝑎𝑛𝑑 𝑈− = 𝜃− 𝑓 𝑘− − 𝑡− . IC PC are same of price discrimination but q=f(k).
28

29. Financial Contracts Solution
 max 𝑣 𝜃−
𝐹 𝑘−
− 𝑅𝑘−
+ 1 − 𝑣 𝜃− 𝑓(𝑘− − 𝑅𝑘−) − (𝑣𝑈−
+ 1 − 𝑣 𝑈−)
 𝑈− ≥ 𝑈− + Δ𝜃 𝑓(𝑘− binding
 𝑈− ≥ 0 binding
 No capital distortion wrt to FB for high productivity type:
 𝑘−
𝑠𝑏 = 𝑘−∗
and 𝜃−
𝑓′
𝑘∗
= 𝑅 return on capital is equal to risk free rate.
 Downward distortion in the size of the loan given to a low productivity borrower wrt to
FB we have 𝑘−
𝑠𝑏
< 𝑘−
∗
where
 [𝜃− −
𝑣
1−𝑣
Δ𝜃]𝑓′(𝑘) = 𝑅 𝑎𝑛𝑑 𝜃_ − 𝑓′(𝑘−
∗ ) = 𝑅
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30. Source
 The Theory of Incentives, Laffont and Martimort (2014) pp. 32-53; 51-53; 72-76.
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