1. Static Games of Incomplete Information.

2. Mechanism design Typically a 3-step game of incomplete info Step 1: Principal designs mechanism/contract Step 2: Agents accept/reject the mechanism Step 3: Agents that have accepted, play the game specified by mechanism Constant theme: Incomplete information and binding individual rationality constraints prevent efficient outcomes

3. Nonlinear pricing A monopolist produces good at marginal cost c and sells quantity q Consumer transfers T to seller and has utility u 1 (q, T, θ)= θV(q)-T, V(0)=0, V / >0, V // <0 θ is private knowledge for buyer Seller knows that θ= w.p. and θ= w.p. The game: 1. Seller offers tariff T(q): specifies a price for qty q 2. Consumer accepts/rejects If seller knows θ, she will charge T= θV(q), her profit, θV(q)-cq. This is maximized at some q given by θV / (q)=c

4. Nonlinear pricing Let be bundle for type and for type Seller’s expected profit: Seller faces two constraints: 1. Individual Rationality (IR): Consumer should be willing to purchase 2. Incentive Compatibility (IC): Consumer should consume the bundle intended for his type IR 1 : ; and IR 2 : IC 1 : ; and IC 2 : First step: To show that only IR 1 and IC 2 are binding

5. Nonlinear pricing First note: IR 1 and IC 2 imply IR 2 IR 2 can’t be binding unless =0 However, IR 1 must bind. Else seller can increase by same amount and increase revenue Also, IC 2 must be binding, else seller can increase, satisfy all constraints and increase revenue The high-type’s indifference curve is always steeper than the low type’s for any allocation This implies that high type consumes more than low type:

6. Nonlinear pricing Eliminating transfers, principal’s objective function is: FOC wrt Check that IC 1 is satisfied Note: Quantity purchased by high-type is optimal Quantity purchased by low-type is sub-optimal Seller sacrifices efficiency for rent-extraction!

7. Auctions Seller has unit of good and there are two bidders Each bidder can have types, with < Corresponding probabilities are and Buyer’s expected probability of getting the good are and payments are The constraints are: IR 1 :; IR 2 : IC 1 : ; IC 2 : What is seller’s optimal contract?

8. Auctions Seller’s expected profit is: Again, IR 1 and IC 2 are binding. The seller’s profit: Also, ex-ante prob of a player getting good, Moreover, Case 1:. The seller sets and Optimal mechanism: Not to sell if both announce low-type; sell to high-type if they announce different types; sell wp ½ to each if both announce high type Case 2:. The seller sets and Optimal mechanism: Sell to high-type if bidders announce different types, and sell wp ½ to each if they both announce high-type or low-type

9. Moral Hazard Consider a Principal and an agent who can exert costly effort, e Let e {0, 1}, with costs: ψ(0)=0, ψ(1)= ψ Agent receives transfer, t, and has utility; U=u(t)- ψ(e), with u / >0, u // <0. Production is stochastic, and production level,, Stochastic influence of effort on production:,

10. Moral Hazard Principal can offer a contract, {t( )}, that depends on observed, random output With two possible outcomes, contract is: if output is and if output is Let Principal’s profit with qty q be S(q) His profit when agent expends effort e=0 is: His profit when agent expends effort e=1 is:

11. Incentive Feasible Contracts Induce positive effort and ensure participation Incentive constraint: Participation constraint:

12. Complete Information Benchmark Complete info or First-Best: Principal observes effort Principal’s problem is: subject to: Using Lagrangian, μ, and from FOCs we have, From the above equations, we have that: Thus, Agent obtains full insurance! The optimal transfer is: t * = u -1 (ψ)=h(ψ), where h=u -1

13. First Best Case When there is complete information Principal’s profit from inducing effort e=1: V 1 = If agent exerted 0 effort, principal would earn: V 0 = Inducing effort is optimal for principal if:, where Principal’s First-Best cost of inducing effort is: h(ψ)

14. Second-Best: In terms of transfers Agent is risk-averse Principal’s problem, P, is: (P): subject to:, and First ensure concavity of (P): Let

15. Second-Best: In terms of utilities The Principal’s program can be rewritten in terms of utilities (P / ): Principal’s objective function is concave in because h(.) is convex, and the constraints are linear The KKT conditions are necessary and sufficient

16. Both IR and IC are binding Let λ & μ be Lagrange multipliers for IC & IR The FOCs, upon rearranging terms, are: where, are second-best optimal transfers From these,, so IR is binding Also,, so IC is binding

17. Second-Best Solution The variables (, λ, μ ) are solved simultaneously from two FOCs, IC and IR The second-best optimal transfers are: : contract does not provide full insurance 2 nd Best cost of inducing effort: C SB = Clearly, for the Principal, C SB > C FB. So Principal induces high effort (e=1) less often than in first-best There is under-provision of effort in the second-best

18. Mechanism design with a single agent Agent’s type with distribution/density Type-contingent allocation is fn. Defn: A decision function is implementable if there exists a transfer t(.) such that allocation y(.) is incentive-compatible, i.e. Theorem: A piecewise C 1 decision fn x(.) is implementable only if whenever and x is differentiable at θ

19. Mechanism design with a single agent Sketch of proof: Type θ announces to maximize The FOC and SOC are Totally differentiating the first equation, The (local) SOC becomesor, Rewrite the FOC we get, Eliminating, dt/dθ,

20. Mechanism design with a single agent The sorting/ single crossing/ constant sign (CS) condition is: Note thatis agent’s marginal rate of substitution between decision k and transfer t Consider x to be output supplied by agent, i.e., Then sorting condition means that the agent’s indifference curve in (x, t) space,, is decreasing in θ If θ 2 > θ 1, y(θ 1 )=(x(θ 1 ), t(θ 1 )), y(θ 2 )=(x(θ 2 ), t(θ 2 )), then y(θ 2 )>y(θ 1 ) Theorem: If decision space is 1-dim and CS holds, then a necessary condition for x(.) to be implementable is that it is monotonic. What about sufficiency?

21. Optimal mechanisms for one agent The assumptions: A1: Reservation utility independent of type A2: Quasi-linear utilities: Principal: u 0 (x, t,θ)= V 0 (x, θ)-t; Agent: u 1 (x, t,θ)= V 1 (x, θ)+t A3: n=1, i.e., decision is 1-dim and CS holds. A4: A5: A6:

22. Optimal mechanisms for one agent The problem: Principal maximizes his expected utility subject to: (IR) u 1 (x(θ), t(θ), θ)≥ =0, for all θ (IC) u 1 (x(θ), t(θ), θ)≥ From A1 & A4, if IR satisfied at, it is satisfied everywhere IR binding at. Thus, Let From Envelope theorem, This implies that,

23. Optimal mechanisms for one agent Further, u 0 = V 0 + V 1 - U 1 ≡ Social surplus-Agent’s utility Principal’s objective function: Since monotonicity is necessary and sufficient for implementability, Principal’s optimization program becomes s.t. x(.) is monotonic

24. Optimal mechanisms We solve the principal’s program ignoring monotonicity The solution to the relaxed program is The principal faces a trade-off between maximizing total surplus (V 0 + V 1 ) and appropriating the agent’s info rent (U 1 ) When is it legit to focus on relaxed program? When solution x * (θ) to above eq is monotonic. Differentiating, When Hazard rate is monotone: