Dunia López-Pintado Universidad Pablo de Olavide and Juan D. Moreno-Ternero Universidad de Málaga and CORE, Université catholique de Louvain (2008-2009: On leave at Universidad Pablo de Olavide) SAE2008; Zaragoza, December 2008
Introduction A recurrent dilemma in team management: Team-based Vs. Individual-based wage schemes
Introduction A recurrent dilemma in team management: Team-based Vs. Individual-based wage schemes We explore such a dilemma in a simple model of production in teams, in which the team members may differ in their effort choices and qualification.
Introduction A recurrent dilemma in team management: Team-based Vs. Individual-based wage schemes We explore such a dilemma in a simple model of production in teams, in which the team members may differ in their effort choices and qualification. We obtain the optimal management strategies under two scenarios: when wages are contingent on the team s success and when they are not.
Introduction A recurrent dilemma in team management: Team-based Vs. Individual-based wage schemes We explore such a dilemma in a simple model of production in teams, in which the team members may differ in their effort choices and qualification. We obtain the optimal management strategies under two scenarios: when wages are contingent on the team s success and when they are not. We show that, despite of promoting incentives, the optimal strategy under the contingent scenario is typically less profitable for the principal.
Introduction Our model and results will also allow us to:
Introduction Our model and results will also allow us to: 1. Make a case against some incentive-enhancing policies as well as team-based wage schemes.
Introduction Our model and results will also allow us to: 1. Make a case against some incentive-enhancing policies as well as team-based wage schemes. 2. Provide rationale for the so-called rich-get-richer hypothesis.
Introduction Our model and results will also allow us to: 1. Make a case against some incentive-enhancing policies as well as team-based wage schemes. 2. Provide rationale for the so-called rich-get-richer hypothesis. 3. Highlight a deep misalignment between designing optimal incentives for a team and treating its members impartially.
The model There is a project involving n activities performed by n agents of a team
The model There is a project involving n activities performed by n agents of a team Each agent decides simultaneously whether to exert effort or not towards the performance of her activity
The model There is a project involving n activities performed by n agents of a team Each agent decides simultaneously whether to exert effort or not towards the performance of her activity The cost of exerting effort for agent i is c i We assume, without loss of generality, that c 1 c 2 c n
The model There is a project involving n activities performed by n agents of a team Each agent decides simultaneously whether to exert effort or not towards the performance of her activity The cost of exerting effort for agent i is c i We assume, without loss of generality, that c 1 c 2 c n Agents are risk neutral (i.e., they invest if and only if their expected benefits are non-negative) and have limited liability
The model There is a project involving n activities performed by n agents of a team Each agent decides simultaneously whether to exert effort or not towards the performance of her activity The cost of exerting effort for agent i is c i We assume, without loss of generality, that c 1 c 2 c n Agents are risk neutral (i.e., they invest if and only if their expected benefits are non-negative) and have limited liability The project s technology is an increasing function p : {0, 1,..., n} [0, 1] specifying the probability of success for any given number of agents exerting effort
The model There is a project involving n activities performed by n agents of a team Each agent decides simultaneously whether to exert effort or not towards the performance of her activity The cost of exerting effort for agent i is c i We assume, without loss of generality, that c 1 c 2 c n Agents are risk neutral (i.e., they invest if and only if their expected benefits are non-negative) and have limited liability The project s technology is an increasing function p : {0, 1,..., n} [0, 1] specifying the probability of success for any given number of agents exerting effort Agents costs (skills) and their effort decisions are observable
The model The principal: designs the team s wage scheme ω = (ω1, ω 2,..., ω n ) earns β > 0 if the project is successful is risk neutral and thus maximizes her expected benefits
The model The principal: designs the team s wage scheme ω = (ω1, ω 2,..., ω n ) earns β > 0 if the project is successful is risk neutral and thus maximizes her expected benefits Since effort is observable, ω i = 0 for each undeserving agent i
The model The principal: designs the team s wage scheme ω = (ω1, ω 2,..., ω n ) earns β > 0 if the project is successful is risk neutral and thus maximizes her expected benefits Since effort is observable, ω i = 0 for each undeserving agent i Two management options for deserving agents
The model The principal: designs the team s wage scheme ω = (ω1, ω 2,..., ω n ) earns β > 0 if the project is successful is risk neutral and thus maximizes her expected benefits Since effort is observable, ω i = 0 for each undeserving agent i Two management options for deserving agents Option 1: to commit to a positive wage for each of them
The model The principal: designs the team s wage scheme ω = (ω1, ω 2,..., ω n ) earns β > 0 if the project is successful is risk neutral and thus maximizes her expected benefits Since effort is observable, ω i = 0 for each undeserving agent i Two management options for deserving agents Option 1: to commit to a positive wage for each of them Option 2: to commit to a positive wage for each of them, but only provided the project is successful
Results Theorem The following statements hold:
Results Theorem The following statements hold: 1. If wages are contingent on the project s success, the principal solves ( ) k c i max p(k) β. k p(i) i=1
Results Theorem The following statements hold: 1. If wages are contingent on the project s success, the principal solves ( ) k c i max p(k) β. k p(i) i=1 2. If wages are not contingent on the project s success the principal solves k max p(k) β c i. k i=1
Proof of the theorem We focus on the contingent case, looking first for an investment-inducing mechanism.
Proof of the theorem We focus on the contingent case, looking first for an investment-inducing mechanism. Since the profile with all agents shirking cannot be a Nash equilibrium, there must exist an agent i 1 that prefers to invest, i.e., ω i1 p(1) c i1
Proof of the theorem We focus on the contingent case, looking first for an investment-inducing mechanism. Since the profile with all agents shirking cannot be a Nash equilibrium, there must exist an agent i 1 that prefers to invest, i.e., ω i1 p(1) c i1 Since the profile with all agents, except i 1, shirking cannot be a Nash equilibrium either, there must exist an agent i 2 i 1 that prefers to exert effort, i.e., ω i2 p(2) c i2
Proof of the theorem If we continue with this argument, we find that the contingent investment-inducing mechanism must satisfy that ( ) c1 (ω 1,..., ω n ) p(π(1)),..., c n p(π(n)) where π is a given permutation.
Proof of the theorem If we continue with this argument, we find that the contingent investment-inducing mechanism must satisfy that ( ) c1 (ω 1,..., ω n ) p(π(1)),..., c n p(π(n)) where π is a given permutation. Notice that the cheapest wage scheme to reach this goal is benefit ( ) c1 (ω 1,..., ω n ) = p(1),..., c n p(n)
Results If we compare both objective functions, it is straightforward to show that ( ) k c i k p(k) β p(k) β c i, p(i) i=1 with a strict inequality for any k > 1 i=1
Results If we compare both objective functions, it is straightforward to show that ( ) k c i k p(k) β p(k) β c i, p(i) i=1 with a strict inequality for any k > 1 Corollary We have the following: i=1
Results If we compare both objective functions, it is straightforward to show that ( ) k c i k p(k) β p(k) β c i, p(i) i=1 with a strict inequality for any k > 1 Corollary We have the following: 1. The principal gets higher expected benefits when wages are not contingent on the project s success i=1
Results If we compare both objective functions, it is straightforward to show that ( ) k c i k p(k) β p(k) β c i, p(i) i=1 with a strict inequality for any k > 1 Corollary We have the following: 1. The principal gets higher expected benefits when wages are not contingent on the project s success i=1 2. Agents get higher expected benefits when wages are contingent on the project s success
Further insights Statement 1 of the corollary says that a principal without liquidity constraints typically obtains higher benefits than a principal with liquidity constraints
Further insights Statement 1 of the corollary says that a principal without liquidity constraints typically obtains higher benefits than a principal with liquidity constraints Thus, our result provides rationale for the so-called rich-get-richer hypothesis
Further insights Statement 1 of the corollary says that a principal without liquidity constraints typically obtains higher benefits than a principal with liquidity constraints Thus, our result provides rationale for the so-called rich-get-richer hypothesis Statement 2 says that agents prefer a pro-incentive (and risky) contingent wage scheme rather than a secure wage
Further insights Statement 1 of the corollary says that a principal without liquidity constraints typically obtains higher benefits than a principal with liquidity constraints Thus, our result provides rationale for the so-called rich-get-richer hypothesis Statement 2 says that agents prefer a pro-incentive (and risky) contingent wage scheme rather than a secure wage Thus, team workers do not prefer principals without liquidity constraints: they would provide them with lower expected benefits as a result of avoiding contingent wage schemes
Further insights The proof of the theorem provides more information regarding agents (expected) wages:
Further insights The proof of the theorem provides more information regarding agents (expected) wages: 1. The optimal non-contingent wage scheme is, ex-ante, minimally egalitarian and, ex-post, extremely egalitarian: agents with the same (different) skills receive equal (different) wages; and all agents end up obtaining the same benefits, despite having different skills
Further insights The proof of the theorem provides more information regarding agents (expected) wages: 1. The optimal non-contingent wage scheme is, ex-ante, minimally egalitarian and, ex-post, extremely egalitarian: agents with the same (different) skills receive equal (different) wages; and all agents end up obtaining the same benefits, despite having different skills 2. The optimal contingent wage scheme is, ex-ante and ex-post, inegalitarian: agents with the same skills might well receive different wages and also end up obtaining different benefits
Results Corollary The optimal size of the team when the principal uses the contingent management option is never higher than when using the non-contingent management option.
Results Corollary The optimal size of the team when the principal uses the contingent management option is never higher than when using the non-contingent management option. The corollary says that if the team is managed under a profit-sharing plan, less agents are expected to exert effort.
Results Corollary The optimal size of the team when the principal uses the contingent management option is never higher than when using the non-contingent management option. The corollary says that if the team is managed under a profit-sharing plan, less agents are expected to exert effort. Thus, if the principal values per se that members of the team exert effort, this shows an additional advantage of the non-contingent management strategy.
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes The existence of wage-equity constraints
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes The existence of wage-equity constraints Risk aversion*
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes The existence of wage-equity constraints Risk aversion* The results are not robust to the assumptions of:
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes The existence of wage-equity constraints Risk aversion* The results are not robust to the assumptions of: The existence of hierarchical organizations
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes The existence of wage-equity constraints Risk aversion* The results are not robust to the assumptions of: The existence of hierarchical organizations The existence of optimistic principals
Further insights The results are robust to the assumptions of: The existence of semi-contingent wage schemes The existence of wage-equity constraints Risk aversion* The results are not robust to the assumptions of: The existence of hierarchical organizations The existence of optimistic principals The existence of more general contracts