ECON INFORMATION ECONOMICS. Lecture 2 February 2016 Slides based on Laffont & Martimort UiO, Spring 2016

Transcription

1 ECON INFORMATION ECONOMICS Lecture 2 February 2016 Slides based on Laffont & Martimort UiO, Spring 2016

2 Asymmetric information A knows his own cost-type θ P does not know the cost-type θ of A What if P offers to A the possibility to choose between the equilibrium First- Best contract derived before? A can freely choose between (q, θ) and (q, θ) Look at the previous graph! An high-cost (θ) A will never pick the contract (q, θ) A low-cost (θ) A will now pick (q, θ), that is the contract that was designed for the other guy and he makes a positive profit (information rent)! A low-cost θ has the incentive to mimick (pretend to be) the high-cost θ The menu of contracts { (q, θ ); (q, θ ) } is not incentive compatible ECON Information Spring /15

3 Information rents Reconsider the efficient type mimicking an inefficient type (that is, pretending to be inefficient by lying) t θq = t θq + θq θq = U + θq U = t θq is what the inefficient earns under the contract designed for him U = 0 under complete information (it s the participation constraint) By mimicking, the efficient type earns a profit θq > 0 θq > 0 is an information rent: it arises from the informational asymmetry ECON Information Spring /15

4 Incentive compatibility and partecipation Incentive Compatibility Constraints (ICCs) (ICC): t θq t θq (ICC): t θq t θq Participation Constraints (PCs) (P C): t θq 0 (P C): t θq 0 A menu of contract is incentive compatible if the ICCs are satisfied A menu of contract is incentive feasible if both ICCs and PCs are satisfied Monotonicity: ICC + ICC q q ECON Information Spring /15

5 P s problem under asymmtric information s.t. The solution to this problem is the optimal contract { max (q,t),(q,t) } ν(s(q) t) + (1 ν)(s(q) t) t θq t θq t θq t θq t θq 0 t θq 0 ECON Information Spring /15

6 Reformulating P s problem under asymmetric information Define as before U = t θq and U = t θq The problem: { max (q,t),(q,t) can be re-written as: } ν(s(q) t) + (1 ν)(s(q) t) { max (q,u),(q,u) } Expected Allocative Efficiency {}}{ ν(s(q) θq) + (1 ν)(s(q) θq) [νu + (1 ν)u] }{{} Expected Information Rent Trade-off between efficiency and rent extraction ECON Information Spring /15

7 Reformulating the ICCs and the PCs t θq t θq U U + θq t θq t θq U U θq t θq 0 U 0 t θq 0 U 0 ECON Information Spring /15

8 P s problem under asymmtric information The solution to this problem is the optimal contract { s.t. max (q,u),(q,u) } Expected Allocative Efficiency {}}{ ν(s(q) θq) + (1 ν)(s(q) θq) [νu + (1 ν)u] }{{} Expected Information Rent U U + θq U U θq (1) (2) U 0 (3) U 0 (4) ECON Information Spring /15

9 ICC : ICC : P C : U 0 P C : U 0 U U + θq U U θq 1. P C and ICC imply P C drop P C 2. Guess: ICC always holds (do your remember the graph?) temporarely drop ICC, and check that the solution satisfies ICC ex-post 3. P C must be binding (i.e. =0) at the optimum proof: if U = ɛ > 0, then P can increase the objective function by ɛ by setting U = 0 4. also ICC (that now reads U θq ) must be binding proof: if U = θq + ɛ > θq, then P can increase the objective function by ɛ by setting U = θq ECON Information Spring /15

10 The final formulation of P s problem { max (q,u),(q,u) } ICC : Expected Allocative Efficiency {}}{ ν(s(q) θq) + (1 ν)(s(q) θq) [νu + (1 ν)u] }{{} Expected Information Rent U = θq P C : U = 0 Plug the constraints into the objective function to get: { max} q,q Expected Allocative Efficiency {}}{ ν(s(q) θq) + (1 ν)(s(q) θq) } ν θq {{} Expected Information Rent FOCq : FOCq : S (q SB ) = θ (1 ν)(s (q SB θ)) = ν θ ECON Information Spring /15

11 FOCq : FOCq : S (q SB ) = θ (1 ν)(s (q SB θ)) = ν θ Recall: we still need to check that ICC holds. First rewrite FOCq: S (q SB ν ) = (1 ν) θ + θ ν = (1 ν) θ ν (1 ν) θ + θ 1 = (1 ν) θ ν (1 ν) θ We can prove that q SB < q SB by checking that S (q SB ) > S (q SB ) S (q SB ) = 1 (1 ν) θ ν (1 ν) θ >θ = S (q SB ) 1 (1 ν) θ > 1 (1 ν) θ ECON Information Spring /15

12 Now we can ex-post check that ICC holds at the SB optimum: ICC : P C : U =0 ICC : U = θq SB U U θq SB 0 θq SB θq SB 0 θ [q SB q SB ] }{{} < 0 as in previous slide ECON Information Spring /15

13 A close look at the SB equilibrium FOCq : FOCq : S (q SB ) = θ (1 ν)(s (q SB ) θ) = ν θ The F OC q is the same as in the complete info case: q SB = q No output distortion for the efficient type From the F OC q : S (q SB ) = ν 1 ν θ + θ > θ = S (q ) Downward output distortion for the inefficient type: q SB < q The efficient type enjoys a positive information rent The inefficient time enjoys zero information rent In other words, compared to the complete info case, the P manages to extract all the rent only from the inefficient type, but not from the efficient type ECON Information Spring /15

14 Incentive compatibility and partecipation: special cases Pooling (or bunching) contracts P offers only one contract ICCs are always and trivially satisfied (tell me why) This solves the incentives problem but it is not very flexible Both types always participate Shutting down the least efficient type P makes sure the low type never participates P offers two contracts: (q, t) = (0, 0) and (q s, t s ) Set (q s, t s ) such that: t s θq s = 0 t s θq s < 0 Only the efficient type participates, while the inefficient never does so ECON Information Spring /15

15 A special case: shutdown policy Shutdown occurs when the F OC q : (1 ν)(s (q SB ) θ) = ν θ has no positive solution Since q SB can not be negative, it must be set to 0 It is trivially incentive compatible: no incentive for the efficient A to mimick The contract must ensure participation of the efficient type: binding P C The (unique) shutdown contract is therefore the FB contract (q, t ) More generally, the shutdown policy is optimal when ν(s(q ) θq ) ν(s(q SB ) θq SB θq SB ) + (1 ν)(s(q SB ) θq SB ) which we can re-write, using q = q SB ν θq SB }{{} Expected info rent (1 ν)(s(q SB ) θq SB ) }{{} Expected benefit from participation of θ ECON Information Spring /15