Principal-Agent model: Hidden Information Laont-Martimort, ch 2,3 Agent is the producer who will produce q units of the good for the principal rm-regulator or worker - employer Principal's utility is V (q; t) = S(q) t Agent's high marginal cost 2 f, g
Probability of low marginal cost is v C(q; ) = q + F = Agent has outside option { reservation utility { normalized to zero. Game as follows: 1. Nature chooses type of agent. 2. A observes his own type, principal does not
3. P oers contract 4. A accepts or rejects contract 5. If A accepts, contract is executed. Contract oered at interim stage, when there is already asymmetric information, P has full commitment. t(q): q is veriable, so principal can promise any transfer First Best Benchmark: Complete information Output levels q; q :
S 0 (q) = S 0 (q) = t = q + F S(q) t = q + F Second Best: Asymmetric Information
Contract is a menu (q; t; q; t) First contract for low cost agent. Second for high cost agent Menu of contracts is incentive compatible if the high cost type weakly prefers (q; t), and the low cost type weakly prefers (q; t) t q t q (ICH) t q t q (ICL)
Participation constraint: Each type of agent must nd it optimal to accept the contract t q 0 (PCH) t q 0 (PCL) An incentive feasible contract satises IC and PC for both types of agent. From ICH and ICL,
t t (q q) t t (q q) This implies ( )(q q) 0 q q
ICL + ICH ) monotonicity constraint Any pair q; q that is implementable must satisfy this constraint. Conversely, for any pair q q; we can nd transfers that satisfy ICL and ICH. Principal's problem: n max v[s(q) t] + (1 v)[s(q) t] o q;t;q;t subject to ICL, PCL, ICH, PCH.
Agent's information rent is U = t q U = t q Rewrite the P's max problem as:
max U;q; U;q V = n v[s(q) q] + (1 v)[s(q) q] o n vu + (1 v) U o U U + q (ICL') U U q (ICH') U 0 (PCH')
U 0 (PCL')
If PCH' is satifsed, then PCL' is satised. If ICL' is satised and binds, then ICH' is satised. So relevant constraints are ICL and PCH Both constraints must be binding at optimum. max q;q V = n v[s(q) q] + (1 v)[s(q) q] o fv qg S 0 (q ) =
(1 v)[s 0 (q ) ] = v Under asymmetric information, the optimal contract has: 1) no distortion for the ecient type. 2) downward output distortion for the high cost type 3) informational rent for the ecient type = q 4) no informational rent for the high cost type
Continuum of types Agent's high marginal cost 2 [, ] Distributed with density function f(:) and cdf F (:) C(q; ) = q + F Agent has outside option { reservation utility { normalized to zero. Game as follows:
1. Nature chooses type of agent. 2. A observes his own type, principal does not 3. P oers contract: direct revelation mechanism, fq(~ ); t( ~ )g 4. A accepts or rejects contract 5. If A accepts, he announces type. Contract is executed. Contract oered at interim stage, when there is already asymmetric information, P has full commitment. t(q): q is veriable, so principal can promise any transfer
Menu of contracts is incentive compatible if for all ; ~ : t() q() t(~ ) q( ~ ) (IC) Write IC for ~ t(~ ) ~ q( ~ ) t() ~ q() Adding these together we get: ( ~ )[q() q( ~ )] 0
Monotonicity requirement: q() is decreasing (weakly) Hence q() is dierentiable almost everywhere (as is t()) Given a direct revelation mechanism, agent of type chooses announcement ^ to maximize u(; ^) = t(^) q(^) @u @^ = t 0 (^) q 0 (^) This must equal zero at ^ = ; i.e. truth telling must be optimal
Local Incentive Compatibility : rst order condition t 0 () q 0 () = 0 Second order condition: t 00 () q 00 () 0 Second order condition will be satised if monotonicity is satised:
q 0 () 0 (totally dierentiate rst order condition) t 00 () q 00 () q 0 () = 0 Global incentive compatibility t() t ~ = Z ~ t 0 (x)dx
= Z ~ xq 0 (x)dx Z = [xq(x)] ~ q(x)dx ~ t() t ~ = q() ~ q( ~ ) Z ~ q(x)dx t() q() = t ~ q(~ ) + ( ~ )q( ~ ) Z ~ q(x)dx ( ~ )q( ~ ) Z ~ q(x)dx 0
LIC rst order condition + monotonicity, global incentive compatibility Participation constraint: for all U() = t() q() 0 U 0 () = t 0 () q 0 () q() = q() from LIC Principal's problem:
Z max q();u() [S(q()) q() U()]f()d subject to U 0 () = q() q 0 () 0 U() 0
No rent for highest cost type, i.e. U( ) = 0 Ignore monotonicity constraint. U() = U( ) Z = Z q(x)dx U 0 (x)dx max q() Z ( S(q()) q() Z q(x)dx ) f()d
Dene Q() = R q(x)dx Z Q()f()d = [Q()F ()] Z q()f ()d max q() Z ( S(q()) " + F () f() # q() ) f()d S 0 (q()) = + F () f() Check that this satises monotonicity condition
Monotone hazard rate property hazard rate is h() = f() 1 F () If this is increasing, then q() will be increasing. Under asymmetric information, the optimal contract has: 1) no distortion for the most ecient type. 2) downward output distortion for all other types 3) informational rent for all types except most inecient type