8.6 Joint Production by Many Agents: The Holmstrom Teams Model The existence of a group of agents results in destroying the effectiveness of the individual risk-sharing contracts, because observed output is a joint function of the unobserved effort of many agents. The actions of a group of players produce a joint output , and each player wishes that the others would carry out the costly actions. A team is a group of agents who independently choose effort levels that result in a single output for the entire group.
Teams ð Players a principal and agents r n ð The order of play 1 The principal offers a contract to each agent of the form i w q ( ), i where is total output. q 2 The agents decide whether or not to accept the contract. œ 3 The agents simultaneously pick effort levels , ( e i 1, . . . , ). n i 4 Output is ( q e , . . . , e ). 1 n
Payoffs ð r If any agent rejects the contract, all payoffs equal zero. r Otherwise, n 1 principal œ q w i œ 1 i and w ww 1 i œ w ( ), where v e v 0 and v 0. i i i i i ð The principal can observe output. ð The team's problem is cooperation between agents.
Efficient contracts e * ð Denote the efficient vector of actions by . An efficient contract is ð * œ ( ) if ( ) (8.9) w q b q q e i i * 0 if q q e ( ), n * * œ where b q e ( ) and b v e ( ). i i i i œ 1 i ð The teams model gives one reason to have a principal : he is the residual claimant who keeps the forfeited output.
Budget balancing and Proposition 8.1 ð The budget-balancing constraint The sum of the wages exactly equal the output. r ð If there is a budget-balancing constraint, no differentiable wage contract w q ( ) generates i an efficient Nash equilibrium. Agent 's problem is i r Maximize w q e ( ( )) ( ). v e i i i e i His first-order condition is Î ` Î` Î œ ( dw dq ) ( q e ) dv de 0. i i i i
The Pareto optimum solves r n Maximize ( ) q e v e ( ). i i , . . . , e e 1 n œ i 1 The first-order condition is that the marginal dollar contribution equal the marginal disutility of effort: ` Î` Î œ q e dv de 0. i i i i Î Á 1 r dw dq Under budget balancing, not every agent can receive the entire marginal increase in output.
Because each agent bears the entire burden of his marginal effort r and only part of the benefit, the contract does not achieve the first-best. Without budget balancing, if the agent shirked a little he would gain the entire leisure benefit from shirking, but he would lose his entire wage under the optimal contract in equation (8.9).
With budget balancing and a linear utility function, the Pareto optimum maximizes the sum of utilities. ð A Pareto efficient allocation is one where consumer 1 is as well-off as possible given consumer 2's level of utility. _ r Fix the utility of consumer 2 at u 2 . ð Maximize w q e ( ( )) v e ( ) 1 1 1 e , e 1 2 subject to _ w q e ( ( )) v e ( ) u 2 2 2 2 and œ w q e ( ( )) w q e ( ( )) ( ) q e 1 2
( ( )) ( ) ð Maximize w q e v e 1 1 1 e , e 1 2 subject to _ œ q e ( ) v e ( ) u w q e ( ( )) 2 2 2 1 _ ð Maximize ( ) q e ( v e ( ) v e ( )) u 1 1 2 2 2 e , e 1 2
Discontinuities in Public Good Payoffs There is a free rider problem ð if several players each pick a level of effort which increases the level of some public good whose benefits they share. r Noncooperatively, they choose effort levels lower than if they could make binding promises .
Consider a situation in which identical risk-neutral players produce ð n a public good by expending their effort. r Let represent player 's effort level, and e i i let ( q e , . . . , e ) the amount of the public good produced, 1 n where is a q continuous function. r Player 's problem is i Maximize ( q e , . . . , e ) . e 1 n i e i His first-order condition is ` Î` œ q e i 1 0.
e * r The greater , first-best -tuple vector of effort levels n is characterized by n ` Î` œ ( q e ) 1 0. i œ 1 i * If the function were discontinuous at ð q e * * œ œ (for example, q 0 if e e and q e if e e for any ), i i i i i i e * the strategy profile could be a Nash equilibrium . e * ð The first-best can be achieved because the discontinuity at makes every player the marginal, decisive player. r If he shirks a little, output falls drastically and with certainty.
Either of the following two modifications restores ð the free rider problem and induces shirking : r Let be a function not only of effort but of q random noise . Nature moves after the players. Uncertainty makes the expected output a continuous function of effort. Let players have incomplete information about the critical value. r e * Nature moves before the players and chooses . Incomplete information makes the estimated output a continuous function of effort.
The discontinuity phenomenon is common. Examples include: ð Effort in teams (Holmstrom [1982], Rasmusen [1987]) ð Entry deterrence by an oligopoly (Bernheim [1984b], Waldman [1987]) Output in oligopolies with trigger strategies ð (Porter [1983a]) ð Patent races ð Tendering shares in a takeover (Grossman & Hart [1980]) Preferences for levels of a public good. ð
Pareto optimum ( , ) ð Maximize q e e e 1 2 1 e , e 1 2 subject to _ œ q e ( , e ) e u 1 2 2 2 ð To solve the maximization problem, we set up the Lagrangian function: _ œ - L ( q e , e ) e { ( q e , e ) e u }. 1 2 1 1 2 2 2
We have the following set of simultaneous equations: _ ` Î` - œ œ L { ( q e , e ) e u } 0 1 2 2 2 ` Î` œ ` Î` - ` Î` œ L e q e 1 q e 0 (A1) 1 1 1 ` Î` œ ` Î` - ` Î` œ L e q e ( q e 1) 0. (A2) 2 2 2 Using expressions (A1) and (A2), we obtain 2 - ` Î` œ - (1 ) ( q e ) 1 , i œ 1 i which leads to 2 ` Î` œ ( q e ) 1 0. i œ 1 i
8.7 The Multitask Agency Problem Holmstrom and Milgrom (1991) ð Often the principal wants the agent to split his time among several tasks , each with a separate output, rather than just working on one of them. ð If the principal uses one of the incentive contracts to incentivize just one of the tasks, this "high-powered incentive" can result in the agent completely neglecting his other tasks and leave the principal worse off than under a flat wage.
Multitasking I: Two Tasks, No Leisure ð Players a principal and an agent r ð The order of play 1 The principal offers the agent either an incentive contract of the form ( w q 1 ) or a monitoring contract that pays under which he pays the agent m m 1 if he observes the agent working on Task 1 and m 2 if he observes the agent working on Task 2.
2 The agent decides whether or not to accept the contract. 3 The agent picks effort levels e and e for the two tasks 1 2 œ 1 such that , e e 1 2 where 1 denotes the total time available. 4 Outputs are q e ( ) and q e ( ), 1 1 2 2 Î Î where dq de 0 and dq de 0 1 1 2 2 but we do not require decreasing returns to effort.
Payoffs ð r If the agent rejects the contract, all payoffs equal zero. r Otherwise, 1 principal œ " q q m w C 1 2 and 2 2 1 agent œ m w e e , 1 2 _ where , the cost of monitoring, is if a monitoring contract is C C used and zero otherwise. r " is a measure of the relative value of Task 2. ð The principal can observe the output from one of the agent's tasks ( q 1 ) but not from the other ( q 2 ).
The first best can be found by choosing and e e 1 2 œ (subject to e e 1) and C to maximize the sum of the payoffs. 1 2 1 principal œ " ð Maximize q e ( ) q e ( ) m w C 1 1 2 2 e , e , C 1 2 subject to _ 2 2 1 agent œ œ m w e e U 0 1 2 and œ e e 1 1 2 _ 1 1 ð Maximize U principal agent e , e , C 1 2 subject to œ e e 1 1 2
The first-best levels of the variables ð * œ 0 r C * œ Î " Î r e 0.5 0.25 { dq de ( dq de )} (8.19) 1 1 2 2 1 * œ Î " Î r e 0.5 0.25 { dq de ( dq de )} 1 1 2 2 2 * * ´ r q ( q e ) i i i r Define the minimum wage payment that would induce the agent to accept a contract requiring the first-best effort levels as * * 2 * 2 ´ w ( e ) ( e ) . 1 2
Can an incentive contract achieve the first best? A profit-maximizing flat-wage contract ð o o o œ r w q ( ) w or the monitoring contract { w , w } 1 o o œ œ r The agent chooses e e 0.5. 1 2 w o œ 0.5 satisfies the participation constraint. r
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