Delegation with Endogenous States Dino Gerardi Lucas Maestri - PowerPoint PPT Presentation

Delegation with Endogenous States Dino Gerardi Lucas Maestri Ignacio Monzón (Collegio Carlo Alberto) (FGV EPGE) (Collegio Carlo Alberto) University of Bonn - October 23rd, 2019

Introduction Delegation � Delegation problems are widespread: � A party with authority to make a decision ( Principal ) � must rely on a better informed party ( Agent ) � Should the principal give ‡exibility to the agent, or instead restrict what the agent can choose? � Some examples: � CEO selects feasible projects Manager (better informed about their pro…tability) chooses one � Regulator restricts the prices that a monopolist (better informed about costs) can charge

Introduction Moral hazard � Before choosing an action, agent can exert e¤ort and a¤ect outcomes � E¤ort is typically unobservable � Agent cannot fully control outcomes � Examples: � Manager’s e¤ort a¤ects potential pro…ts of various projects � Monopolist can adopt practices that reduce production costs

Introduction Goal of the paper � How can a principal incentivize the agent to both exert e¤ort and choose appropriate actions? � Principal chooses a delegation set � Cares about e¤ort and actions � We characterize the optimal delegation set � With aligned and misaligned preferences � The optimal delegation set has a simple form: actions below a threshold are excluded

Introduction Closely related literature � Delegation with misaligned preferences, no moral hazard: � Holmström (1977, 1984) � Alonso and Matouschek (2008) � Amador and Bagwell (2013) � Delegation with Information Acquisition: � Szalay (2005) � Deimen and Szalay (2018)

The model with no bias The model with no bias. Timing Principal selects a delegation set A � R ( A closed) Agent exerts e¤ort e 2 [ 0, e ] at cost c ( e ) Given e¤ort e , the state γ is realized according to c.d.f. F ( γ , e ) Agent observes the state γ and chooses an action a 2 A

The model with no bias Distribution of the state h i The support of the state distribution is Γ = γ , ¯ γ For every e 2 [ 0, e ] and every γ 2 Γ , f ( γ , e ) > 0 F ( � , � ) is smooth F satis…es the (strict) monotone likelihood ratio property (MLRP) : f ( γ 0 , e 0 ) f ( γ , e 0 ) > f ( γ 0 , e ) f ( γ , e ) for all e 0 > e and γ 0 > γ

The model with no bias Payo¤s The parties’ payo¤s are: U P ( a , γ , e ) = u ( a , γ ) + v ( e ) U A ( a , γ , e ) = u ( a , γ ) � c ( e ) Assumptions � v ( � ) : [ 0, e ] ! R is strictly increasing and strictly concave � c ( � ) : [ 0, e ] ! R is strictly increasing and strictly convex

The model with no bias � the common payo¤ component u ( � , � ) is C 2 and satis…es h i � for every γ 2 γ , ¯ γ , u ( � , γ ) is strictly quasiconcave in a and u ( a , γ ) = u ( a � ( γ ) , γ ) = 0 max a h i � Limit condition : for every γ 2 γ , ¯ γ a !� ∞ u ( a , γ ) = lim lim a ! + ∞ u ( a , γ ) = � ∞ � Single crossing condition : for all ( a , γ ) 2 R � Γ ∂ u 2 ( a , γ ) > 0 ∂γ∂ a

The model with no bias Expected payo¤s Given a delegation set A and an e¤ort level e , the parties’ expected payo¤s are: V P ( A , e ) = E [ max a 2 A u ( a , γ ) j e ] + v ( e ) V A ( A , e ) = E [ max a 2 A u ( a , γ ) j e ] � c ( e ) Notice that v ( e ) can be thought as E [ r ( γ ) j e ] where r ( � ) is an increasing function

Results with no bias Floor Delegation De…nition A delegation A set is a ‡oor if A = [ a , + ∞ ) for some a 2 R . The agent’s optimal action when the delegation set is a ‡oor is a 2 [ a , + ∞ ) u ( a , γ ) = max f a , a � ( γ ) g ˆ a ( γ , a ) = arg max

Results with no bias Interval and ‡oor delegation sets Proposition 1 i) Let ˜ A be an optimal delegation set and let ˜ e > 0 be the optimal level of e¤ort. For every γ 2 Γ let ˜ a ( γ ) = max a 2 ˜ A u ( a , γ ) denote the action chosen by the agent when the state is γ . Then the set n � �o a : a = ˜ a ( γ ) for some γ 2 γ , ¯ γ is convex. ii) If there is an optimal delegation set, then there is also an optimal ‡oor delegation set.

Results with no bias Sketch of the proof of Proposition 1 The proof of part i) is by contradiction Assume that Z ¯ Z ¯ γ γ γ u ( a � ( γ ) , γ ) f ( γ , ˜ e ) d γ > γ u ( ˜ a ( γ ) , γ ) f ( γ , ˜ e ) d γ The other case is similar � a � � � i , a � ( γ ) By continuity, there exists a unique a 2 γ such that Z ¯ Z ¯ γ γ γ u ( ˜ a ( γ ) , γ ) f ( γ , ˜ e ) d γ = γ u ( ˆ a ( γ , a ) , γ ) f ( γ , ˜ e ) d γ

Results with no bias Furthermore, quasiconcavity and single crossing of u ( � , � ) γ < ( a � ) � 1 ( a ) such that guarantee that there exists a unique ˆ u ( ˜ a ( γ ) , γ ) > u ( ˆ a ( γ , a ) , γ ) if and only if γ < ˆ γ If the principal adopts the ‡oor delegation set [ a , + ∞ ) , the agent prefers ˜ e to lower levels of e¤ort The di¤erence u ( ˆ a ( γ , a ) , γ ) � u ( ˜ a ( γ ) , γ ) is negative (positive) below (above) ˆ γ Thus, it follows from MLRP that R ¯ γ γ [ u ( ˆ a ( γ , a ) , γ ) � u ( ˜ a ( γ ) , γ )] f ( γ , ˜ e ) d γ > R ¯ γ γ [ u ( ˆ a ( γ , a ) , γ ) � u ( ˜ a ( γ ) , γ )] f ( γ , e ) d γ for every e < ˜ e

Results with no bias e given ˜ From the optimality of ˜ A we have: Z ¯ Z ¯ γ γ e ) > γ u ( ˜ a ( γ ) , γ ) f ( γ , ˜ e ) d γ � c ( ˜ γ u ( ˜ a ( γ ) , γ ) f ( γ , e ) d γ � c ( e ) Combining the two inequalities we obtain: Z ¯ Z ¯ γ γ e ) > γ u ( ˆ a ( γ , a ) , γ ) f ( γ , ˜ e ) d γ � c ( ˜ γ u ( ˆ a ( γ , a ) , γ ) f ( γ , e ) d γ � c ( e ) for every e < ˜ e

Results with no bias If ˜ e < ¯ e and the principal adopts the ‡oor delegation set [ a , + ∞ ) , ˜ e is not optimal (this, again, follows from MLRP) Thus, the optimal e¤ort level e 0 must be larger than ˜ e . We have � ˜ � V A ( a , e 0 ) > V A ( a , ˜ e ) = V A A , ˜ e � ˜ � V P ( a , e 0 ) > V P A , ˜ e If ˜ e = ¯ e , then the agent will continue to choose ¯ e even if the principal adopts the ‡oor delegation set [ a � ε , + ∞ ) for some small ε > 0. Again, the original delegation set ˜ A is not optimal

Results with no bias Existence Proposition 2 There exists an optimal delegation set. We restrict attention to ‡oor delegation sets and show that the principal’s optimization problem admits a solution

Results with no bias Comparative Statics Given the ‡oor delegation set [ a , + ∞ ) , let BR ( a ) denote the set of optimal e¤ort levels. Proposition 3 i) If a > a 0 then e > e 0 for every ( e , e 0 ) 2 BR ( a ) � BR ( a 0 ) . ii) Consider two bene…t functions, v 1 ( � ) and v 2 ( � ) with v 0 1 ( e ) > v 0 2 ( e ) for every e . Let e i be an optimal level of e¤ort for the model in which v = v i for i = 1, 2. Then e 1 > e 2 . i ii) Consider two cost functions, c 1 ( � ) and c 2 ( � ) with 1 ( e ) 6 c 0 c 1 ( 0 ) = c 2 ( 0 ) = 0 and c 0 2 ( e ) for every e . Let V i P , i = 1, 2, denote the principal’s payo¤ of the optimal delegation set P > V 2 when the cost is c i ( � ) . Then V 1 P .

Model with bias The model with bias Quadratic payo¤ function and uniform distributions with shifting support Agent is biased towards some action: u P ( a , γ ) = � ( γ + β � a ) 2 u A ( a , γ ) = � ( γ � a ) 2 β > 0 ( β < 0 ): the principal prefers higher (lower) actions than the agent

Model with bias Consider a simple family of probability distributions When the e¤ort is γ > 0 the state is uniformly distributed in the unit interval [ γ , γ + 1 ] Cost function is quadratic: c ( γ ) = γ 2 2 The bene…t function v ( γ ) is concave

Model with bias The delegation set A and the e¤ort level γ induce expected payo¤s: V P ( A , γ ) = � R γ + 1 γ , A )) 2 d ˜ ( ˜ γ + β � ˆ a ( ˜ γ + v ( γ ) γ V A ( A , γ ) = � R γ + 1 γ , A )) 2 d ˜ γ � γ 2 ( ˜ γ + β � ˆ a ( ˜ γ 2 γ � a ) 2 where ˆ a ( ˜ γ , A ) = arg max a 2 A � ( ˜

Results with bias Necessary conditions for optimal e¤ort Given a delegation set A , the agent solves the following problem: h γ � a ) 2 i R γ + 1 γ � γ 2 max γ > 0 max a 2 ˜ A � ( ˜ d ˜ 2 = γ R γ + 1 γ , A )) 2 d ˜ γ � γ 2 max γ > 0 � ( ˜ γ � ˆ a ( ˜ γ 2 First-order conditions for interior γ : � � �� 2 � � � �� 2 = γ γ , ˜ γ + 1, ˜ γ � ˆ γ + 1 � ˆ a A a A In general, the …rst-order conditions are not su¢cient (the problem is not necessarily concave)

Results with bias Concavity under interval delegation Lemma 1 Suppose that the delegation set is an interval [ a , ¯ a ] for some a � ¯ a . For every γ , let z ( γ ) denote the agent’s expected payo¤ if the e¤ort is γ : � � Z γ + 1 γ � γ 2 z ( γ ) = � max a ] u A ( a , ˜ γ ) d ˜ 2 a 2 [ a ,¯ γ The function z ( � ) is concave.

Results with bias Optimal interval delegation Proposition 4 Let γ > 0 be an optimal level of e¤ort and ˜ A denote the smallest optimal delegation set. Then ˜ A is convex. Moreover, either ˜ A � [ γ , γ + 1 ] or ˜ A = f ¯ a g with ¯ a > γ + 1. To incentive the agent to exert high e¤ort levels the principal may allow only one action: ˜ A = f ¯ a g with ¯ a > γ + 1.