Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 13. Pay for Performance Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 1 / 29
Introduction The moral hazard problem Imagine two individuals, a principal and an agent. The principal (the owner of a company) wants the agent (the manager) to work on a project (maximize the company’s value). A moral hazard problem occurs if the following attributes characterize the principal-agent-relationship. Informational Asymmetry. The principal cannot observe (or evaluate) the agent’s behavior. Con‡ict of Interest. The principal and the agent have di¤erent preferences regarding the set of actions the agent can take. For example, the principal wants the agent to work hard on the project, while the agent would rather like to enjoy perks or increase the size of the company. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 2 / 29
Introduction Many situations are characterized by moral hazard: Principal Agent Employer Employee Creditor Borrower Client Financial Advisor Patient Doctor People Government In this chapter, we ask to what extent monetary incentives (performance pay) can solve the moral hazard problem. We will see that monetary incentives work …ne in some circumstances, while in others, they have negative consequences for the principal. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 3 / 29
Introduction Overview The Owner-Manager Con‡ict The First-Best Contract The Second-Best Contract Extensions Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 4 / 29
The Owner-Manager Con‡ict We build a simple model to analyze the owner-manager con‡ict. Our goal is to …nd an incentive contract that maximizes the owner’s utility under the constraint that the manager exerts a certain level of e¤ort. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 5 / 29
The Owner-Manager Con‡ict We call the owner of the company “principal” P , and the manager “agent” A . The possible pro…t levels (or “states of the world”) are x 2 f x 1 , x 2 , ..., x n g , where x 1 < x 2 < ... < x n . A chooses between to actions, a l and a h , where a l represents low e¤ort and a h high e¤ort. The costs of action for A are given by C ( a h ) = c > C ( a l ) = 0. If A does not work for P , he gets his reservation utility ¯ V . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 6 / 29
The Owner-Manager Con‡ict A ’s action determines the probability distribution over the company’s pro…t x : n a h ) p h = ( p h 1 , p h 2 , ..., p h n ) , p h p h ∑ = i > 0 , i = 1 , a i = 1 n a l ) p l = ( p l 1 , p l 2 , ..., p l n ) , p l p l ∑ a = i > 0 , i = 1 . i = 1 The expected value is larger under high e¤ort: n n ∑ p h ∑ p l i x i > i x i . i = 1 i = 1 P is risk-neutral and has the utility function π = x � w , where w is the wage P pays to A . A is risk-averse and has the utility function V = U ( w ) � C ( a ) with U 0 ( w ) > 0 and U 00 ( w ) < 0. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 7 / 29
The Owner-Manager Con‡ict The sequence of events is as follows: P o¤ers an incentive contract w ( x ) . 1 A decides whether to accept or reject P ’s contract. If A rejects the 2 contract, P gets 0 and A gets ¯ V . Otherwise, the game continuous. A chooses his action a . 3 The pro…t x is realized, P gets x � w ( x ) and A gets 4 U ( w ( x )) � C ( a ) . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 8 / 29
The First-Best Contract Suppose that there is no asymmetric information and P can determine A ’s action in the contract (for example, by imposing a huge …ne on A if he deviates to another action). What incentive contract w ( x ) then maximizes P ’s expected pro…ts? We answer this question by going through two steps. First, we determine for each possible action the optimal incentive contract that implements the action at lowest costs. Second, we choose the action that P would like to implement (given the minimal costs from the …rst step). Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 9 / 29
The First-Best Contract We start with a h . The optimal contract solves the problem n p h ∑ i ( x i � w i ) max w i i = 1 subject to A ’s participation constraint (PC) n p h i U ( w i ) � c � ¯ ∑ V . i = 1 In the optimum, PC must be binding (why?). Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 10 / 29
The First-Best Contract The Lagrangian function is " # n n p h ¯ p h ∑ ∑ L = i ( x i � w i ) � λ V � i U ( w i ) + c . i = 1 i = 1 The …rst-order conditions are ∂ L � p h i + λ p h i U 0 ( w i ) = 0 , = ∂ w i n ∂ L i U ( w i ) � c � ¯ ∑ p h = V = 0 . ∂λ i = 1 Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 11 / 29
The First-Best Contract The …rst-order conditions characterize the optimal contract. From the …rst equality we obtain 1 U 0 ( w i ) = λ for all i 2 f 1 , 2 , ..., n g . w ( a h ) , where ¯ w ( a h ) is A ’s wage if This means w 1 = w 2 = ... = w n = ¯ he chooses a h . The constant wage o¤ers A full insurance! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 12 / 29
The First-Best Contract From the second equality we obtain w ( a h )) � c = ¯ U ( ¯ V . We can rewrite this as w ( a h ) = U � 1 ( ¯ ¯ V + c ) . Hence, A is compensated for his outside option and gets his reservation value! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 13 / 29
The First-Best Contract As with high e¤ort, we can derive the optimal incentive contract that implements low e¤ort a l . Again, we obtain a …xed wage w ( a l ) = U � 1 ( ¯ V ) (why?). P then chooses the contract that o¤ers her the highest expected pro…t: n n p h i x i � U � 1 ( ¯ p l i x i � U � 1 ( ¯ ∑ ∑ V + c ) vs. V ) . i = 1 i = 1 Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 14 / 29
The First-Best Contract The optimal contract pays a …xed wage to A if he chooses the action prescribed in the contract. If he chooses another action, he pays a huge …ne so that deviation does not pay o¤. Note that under the optimal contract, both the allocation of risk ( A has full insurance while the risk-neutral P absorbs all the risk) and e¤ort incentives ( A chooses the optimal action) are e¢cient. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 15 / 29
The Second-Best Contract We now consider the interesting case where P cannot observe A ’s action. The contract no longer can condition on a . How does the optimal incentive contract look like under asymmetric information? Under a …xed wage, A would always choose a l . A contract that implements a h has to pay more to A in a state of the world that is more likely after high e¤ort a h than after low e¤ort a l . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 16 / 29
The Second-Best Contract Again, we proceed in two steps. First, we determine for each possible action the optimal incentive contract that implements the action at lowest costs. Second, we choose the action that maximizes P ’s pro…t. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 17 / 29
The Second-Best Contract We start with a h . P ’s maximization problem is n p h ∑ i ( x i � w i ) max w i i = 1 subject to the participation constraint (PC) n p h i U ( w i ) � c � ¯ ∑ V i = 1 and the incentive compatibility constraint (IC) n n ∑ p h ∑ p l i U ( w i ) � c � i U ( w i ) . i = 1 i = 1 The incentive compatibility constraint ensures that it does not pay o¤ for the agent to accept the contract and then choose action a l . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 18 / 29
The Second-Best Contract The Lagrangian function becomes " # n n p h ¯ p h ∑ ∑ L = i ( x i � w i ) � λ V � i U ( w i ) + c i = 1 i = 1 " # n n p l p h ∑ ∑ � µ i U ( w i ) � i U ( w i ) + c . i = 1 i = 1 Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 19 / 29
The Second-Best Contract The …rst-order conditions are ∂ L = � p h i + λ p h i U 0 ( w i ) � µ ( p l i � p h i ) U 0 ( w i ) = 0 , ∂ w i n ∂ L ∑ p h i U ( w i ) � ¯ ∂λ = V � c = 0 , i = 1 n ∂ L ( p h i � p l ∑ ∂µ = i ) U ( w i ) � c = 0 . i = 1 Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 20 / 29
The Second-Best Contract The …rst-order conditions characterize the optimal contract. We must have � � 1 � p l 1 i U 0 ( w i ) = λ + µ for all i 2 f 1 , 2 , ..., n g . p h i The Lagrange multipliers are positive constants (in a next step, we will explain why). This implies that the wage is not constant, but moves with the likelihood-ratio p l i / p h i . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 13. Pay for Performance 21 / 29
Recommend
More recommend
