Notes onAdverse Selection

Preliminary DraftSummary based on The Theory of Incentives, Laffont and Martimort (2014)

Typos and small mistakes may be present, and they are entirely mine.

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2Overview

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

3Basics 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 () or inefficient(). F is fixed cost.

Spread of uncertainty on agent’s marginal cost: Under adverse selection: uncertainty on agent’s type is exogenous.

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 then q is nonnegative, t can be negative. Variables are observable by a court of law.

Timing of the problem:

A discovers his type

P offers a contract

A accepts of rejects the contract

The contract is executed

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5First Best Production Levels(Complete Information)

Suppose Complete information. Efficient production is obtained equating MR and MC.

These production levels are carried out if their social values are non negative:

and Social value of the efficient is greater than inefficient. _>

W=MR-MC of «efficient agent» is greater than «efficient mimicking inefficient», which in turn is greater than «inefficient».

6FB 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.

Principal offers a «take it or leave it» offer to the agent: If agent is inefficient the principal offers a transfer If agent is efficient the principal offers a transfer Whatever its type agent accepts and makes zero profit; Under complete information delegation is costless for the principal.

7Indifference curves of both types

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Inefficient agents have steeper slopes, since theta is higher

u inefficient u efficient u inefficient2 u efficient2 u inefficient3 u inefficient4

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

8First Best Graph

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.

9Incentive 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.

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

Participation Constraint

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11Special 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 IC of inefficient is

Pro: reduces number of constraints.

Cons: principal gives up production if agent is

inefficient. Excessive Screening costs of subset of types.

12Monotonicity Constraints

Quantity of the efficient agent is higher than inefficient agent both IN SB and FB.

Implementability condition Any pair of outputs which are implementable (quantities

associated to IC contract) must satisfy this condition.

Transfers associated to IC :

13Information 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.

14SB Optimal Contract Max 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.

IC and PC: Utility of the efficient is greater than inefficient and info rent: U_>

U_>0 >0

15Constraints 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.

16SB SOLUTIONS Utility is now information rent for efficient agent and zero for inefficient. New maximization of principal: unique information rent new constraint.

Sb Solution: MR = MC : and quantity sb = quantity fb. Increasing Inefficient output increases efficiency but also information rent and

principal payoff is reduced by : ; at SB principal does not increase or decrease the inefficient quantity. This is

Rent Extraction / Efficiency Trade-Off.

17SB 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.

Only efficient obtains positive information rent

SB TRANSFERS ARE GIVEN BY :efficient gets compensation from FB and info rent. inefficient gets compensation from SB cost (lower).

18SB Graph 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.

19Final Optimal SB Menu of Contract: (Asb, Bsb)

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.

20Shutdown 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

Shutdown policy is optimal when the expeced utiliy of the principal from efficient FB is greater than efficient and inefficient SB.

when

212,10 More General Utility function Instead of writing we write a function C(q,theta). Same IC and PC, agent’s utility:

Optimal Contract is then

since MC of phi is positive reducing inefficient agent’s output reduces efficient agent’s information rent.

No output distortion for efficient MR= MC Downard distortion for inefficient

22Optimal Contracts the optimal contract entails: No output distortion with respect to the first best outcome for the efficient type with A downward distortion for the less efficient type: FB was Only efficient type gets a positive information rent given by

Second Best Transfers are:

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.

23Spence 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 : SM generalizes to 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.

𝜙′ (𝑞)>0 𝑎𝑛𝑑𝐶𝑞𝜃>0 ;𝜙 (𝑞𝑠𝑏 )≥𝜙 (𝑞−𝑠𝑏)

24Non 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 ; 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

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.

25Non Linear Monopoly Pricing Subject to: ; binding ; binding and

Efficient type is the one with highest valuation for the good theta upper bar. Hence and are binding.

No output distortion for high valuation type and

Downward distortion for low valuation agent’s output wrt to FB; and and So unit pricec is different if buyers demand q^-FB or q_SB: price are

nonlinear.

26Quality 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)

IC and PC are rewritten in form of information rents. Instead of u(q) it deals with q directly.

binding binding

27Quality and Price Discrimination - ii Highest valuation agent receives the FB quality

Lowest valuation agent receives the quality below FB. q_sb<q_fb. ; 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.

28Financial 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 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. IC and PC are written as borrower’s information rents . IC PC are same of price discrimination but q=f(k).

29Financial Contracts Solution

binding binding

No capital distortion wrt to FB for high productivity type: and 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 where

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30Source

The Theory of Incentives, Laffont and Martimort (2014) pp. 32-53; 51-53; 72-76.