FS IV 93 - 15
Agency in a Market Setting
William Novshek Lynda Thoman
Purdue University
June 1993
ISSN Nr. 0722 - 6748
Forschungsschwerpunkt Marktprozeß und Unter
nehmensentwicklung (IIMV) Research Unit
Market Processes and
Corporate Development (UM)
Agency in a Market Setting
This paper examines a market in which a continuum of principals and agents interact in a game. Principals offer contracts while agents decide on sets of acceptable contracts. A mechanism from a class satisfying "efficiency," "unbiasedness," and
"continuity" properties then matches principals and agents. With risk neutral agents, when the contribution of principals and agents to the total "gains from trade" in a pairing are additively separable, the equilibria of the game coincide with the competitive equilibria for the market. In particular, all contracts used in Nash equilibrium induce first-best effort levels. Both principals and agents have exogenous opportunities outside this market. In equilibrium, agents have endogenously determined outside opportunities available from employment by another principal, and this may be the binding participation constraint in a principal-agent pairing. The results are extended to special non-separable cases and to the case of identical risk averse agents.
ZUSAMMENFASSUNG
Das Principal-Agent Problem auf einem Markt
Die Arbeit untersucht einen Markt, auf dem ein Kontinuum von Principalen und Agenten an einem Spiel teilnimmt. Principale bieten Verträge an und Agenten entscheiden, welche Verträge für sie akzeptabel sind. Ein Mechanismus mit
"Effizienz-", "Unverzerrtheits-" und "Stetigkeits"-Eigenschaften führt dann Principale und Agenten zusammen. Wenn die Agenten risikoneutral sind und sich der Tauschgewinn einer Paarung additiv separabel aus den Beiträgen der Spieler zusammensetzt, entsprechen die Gleichgewichte dieses Spiels den Marktgleichgewichten bei vollständiger Konkurrenz. Insbesondere führen alle Verträge eines Nash-Gleichgewichtes zu "first-best" Anstrengungen der Agenten.
Sowohl Principale als auch Agenten haben exogene Alternativen außerhalb des Marktes. Im Gleichgewicht dagegen werden die Alternativen eines Agenten durch die Möglichkeit, bei einem anderen Principal beschäftigt zu werden, endogen bestimmt.
Diese können mit der bindenden Teilnahmebedingung für eine Principal-Agenten- Paarung übereinstimmen. Die Ergebnisse werden für spezielle nicht-separable Fälle und für den Fall identisch risikoaverser Agenten verallgemeinert.
1. Introduction
Principal-agent models studying the value of communication, such as Melumad and Reichelstein (1989) and models employing menus of contracts, such as Christensen (1981) and Sappington (1984), often deal with a variety of agent types all of which have the same
alternatives outside the principal-agent pairing. In many instances it would seem natural for the value of an agent's outside alternative to be correlated with the desirability of employing the agent within the principal-agent relationship. For example, if agent type is an ability parameter, then unless the job for the principal uses abilities that are unproductive in alternative uses, a high ability agent should have a high outside alternative.
We consider a competitive model with a continuum of principals and a continuum of agents in which there are two types of outside opportunities for an agent. The agent could refuse to work for any principal, in which case she earns a standard outside alternative expected utility level. Considering any single principal-agent pairing, the agent also has endogenously
determined outside opportunities available by switching to another principal. If the jobs offered by principals use similar abilities, then this endogenously determined outside opportunity will be higher for high ability agents.
Assuming risk neutral agents, since that is standard in much of the literature to which our results are related, we provide a game in which principals offer menus of contracts, while agents decide on sets of acceptable contracts. Then any mechanism satisfying certain "efficiency,"
"unbiasedness," and "continuity" properties is used to match principals, contracts, and agents.
Under certain assumptions we show the Nash equilibria of this game correspond to competitive equilibria. In particular, all contracts used in a Nash equilibrium induce first-best effort levels, and there is no advantage to offering a menu of contracts.
To put our results in context, consider a typical value of communication problem. A single principal must hire one risk neutral agent of unknown type drawn from a pool of types with known distribution. This problem is often interpreted as a single principal facing a given
single agent who has observed some relevant precontract information which is unobservable by the principal. Each type has a fixed outside alternative. If the principal knew the agent's type, the optimal contract for the principal would be to "sell the firm" to the agent by setting t(x), the payment to the agent when outcome x arises, so that x - t(x) is constant (i.e., the principal would receive a fixed payment independent of the outcome) and the agent, using the "first-best" effort level, just obtains her outside alternative utility level. The agent would be induced to work at the "first-best" effort level since she would receive all the benefits of extra effort. When the agent's type is unknown, if the principal offered a single "sell the firm" contract which all types would accept, this would induce first-best effort levels but might yield high rents to "high" type agents. As a consequence, the firm may find it more profitable to offer a menu of non-"sell the firm" contracts (with resulting non-first-best effort levels) and allow the hired agent to specify which contract to implement. This trades off non-optimal effort levels against the rents to
"high" types.
In our model, there is a continuum of agents, possibly all of different types, rather than a single agent with a continuum of possible types. There is also a continuum of principals who are
"competing" to attract agents to their projects. Optimal contracts are determined endogenously in equilibrium. In a Nash equilibrium with endogenously determined outside alternatives for agents, the ability of a principal to exploit a menu of contracts to improve expected profit is eliminated. In the Nash equilibria of our game, all contracts offered by principals and matched with agents will induce first-best effort levels.
It is possible to see in a finite example why competition for agents eliminates the ability of principals to exploit menus of contracts to improve profit. Consider a risk neutral principal and a risk neutral agent of unknown "ability" type from a known distribution. Suppose (as is sometimes done in this literature) that if the agent's type were known, the principal would be able to earn strictly positive expected profits with each different type. If the principal does not hire the agent, he gets zero. Now suppose there are m identical principals, and n agents who are identical except that each has an "ability" type which is an independent draw from the
distribution of possible types. Principals are known, and the distribution from which agents' types are drawn is known. The numbers of principals and agents are known. Principals
announce menus of contracts, and agents announce sets of acceptable contracts. After principals and agents simultaneously and independently make their announcements, an allocation
mechanism satisfying certain assumptions matches principals, contracts, and agents.
Suppose there are more principals than agents. What characteristics must a Nash equilibrium in announcements possess? Clearly no principal earns negative expected profit.
Since at least one principal will be unmatched, in equilibrium no other principal must be earning strictly positive expected profit (or the unmatched principal could offer a slightly more attractive contract). The accepted contracts must be of the "sell the firm" type (i.e., must lead to first-best effort choice) or the unmatched principal could offer a more attractive contract and still earn strictly positive profit. In equilibrium, the advantages of menus of contracts have been eliminated by "competition." Principals earn their "reservation expected utility," zero, by
offering a "sell the firm" contract at price zero. All agent types extract all the gains generated by the match with a principal, and "high" types earn high "information" rents.
What happens if there are more agents than principals? Consider the extreme case with a single principal and a continuum of agents of different types where the distribution of types coincides with the distribution of possible types in the previous finite agent case. We interpret a continuum of agents as modeling a situation in which the realized distribution of agent types is known and matches the distribution from which types are drawn. In this context there is again no role for menus of contracts, and the principal offers a "sell the firm" contract. However, this contract is such that only the highest agent type could be employed and receive her reservation expected utility. Here the principal extracts all the gains from the match. There is no need for a menu of contracts since the best agent type is known to exist. (This case is a limit of the results in McAfee and McMillan (1987). They study the case with a single principal and n agents drawn from the same distribution.)
With a continuum of principals and agents, the Nash equilibria will correspond to
competitive equilibria. All contracts used in equilibrium will induce first-best actions, and there will be no need or use for menus of contracts. In equilibrium, marginal principals and marginal agents will be identified. "Higher" type agents will earn rents beyond the expected payoff to the marginal agent. A principal cannot use a menu of contracts to reduce these rents since they correspond to the endogenously determined reservation utility available to the agent by working for another principal. Rents will depend on the sets of principals and agents. With a given distribution of agents, and identical principals, as we reduce the measure of available principals, fewer agents will obtain contracts in the Nash equilibrium, and the marginal agent will be of a
"higher" type. The corresponding rents to employed agents would be lower than in the situation with a larger measure of available principals. If principals also differ in terms of the project they control, then "higher" type principals (in terms of the productivity of projects they control) will earn rents beyond the expected payoff to the marginal principal.
With a finite number of principals and agents, if there are more agents than principals, then the problem is more complicated since the realized distribution of actual agent types is unknown. Menus of contracts could be used to screen types. This is an interesting problem but not our concern in this paper.1
Our results allowing mechanisms satisfying the three properties of "efficiency,"
"unbiasedness," and "continuity," and menus of general contracts are mainly for a separable case where separability means if we consider contracts which induce first-best effort levels, then the
"contribution" by any principal and any agent to a match can be written in a separable form. It is easy to show competitive equilibrium allocations can be supported as Nash equilibrium
allocations. The converse is not as apparent. We show that if agents are parameterized in a one
1 In our ongoing research we are examining an extension of the McAfee and McMillan framework to more than one principal. Principals offer menus of contracts, agents apply for one contract per principal, each principal offers the job to the most attractive applicant, and agents who have been offered jobs decide which to accept. Additional rounds of job offers and acceptances occur until all principals either have an agent or have rejected all applicants.
dimensional way (e.g., ability level) that satisfies certain assumptions, then for any Nash equilibrium with a positive measure of matches, the Nash equilibrium expected utility levels coincide with competitive equilibrium expected utility levels. When we allow general variation among agents, we show that in any Nash equilibrium in which unmatched players are "telling the truth" in the sense that they announce strategies that are the most attractive to the other side of the market without having any risk of leaving the player worse off with a match than without, the Nash equilibrium allocation of expected utilities coincides with competitive equilibrium expected utility levels. A Nash equilibrium which is not robust to "truth telling" strategies by unmatched players is not particularly attractive. We also examine a special non-separable case and a very special case in which agents are not risk neutral, and show that results similar to the parameterized agent case hold.
Our main conclusion is that when many agents and many principals "compete" in a market, the nature of Nash equilibrium is constrained to match competitive equilibrium. All matched contracts must induce first-best effort levels; agents become residual claimants as an outcome, not as an assumption. There is no advantage to offering a menu of contracts. The binding reservation utility for many agents is determined endogenously by their option to work for other principals. "High" type agents may earn substantial rents in equilibrium.
As noted by Hellwig (1987), there are alternative ways to model "competition" with adverse selection. In Gale (1992) a continuum of principals and a continuum of agents are each divided into a finite number of types. A finite number of contract types are possible and both principals and agents respond to the "prices" set by the Walrasian auctioneer.
Since our interest in this problem arose from a different perspective, we have alternative features we wish to emphasize. We assume a continuum of principals and a continuum of agents but we do not restrict them to a finite number of types. We use a continuum of principals and agents because it is the appropriate framework in which to discuss competition. Each individual is negligible in the aggregate. (In contrast to "competition" among a finite number of
individuals, in which individuals are not negligible.) The continuum also removes some
lumpiness problems that sometimes arise in finite models in which individuals are assumed to act competitively.
Our approach—principals and agents make announcements, the mechanism uses the announcements to decide matches, and matched principals and agents use the corresponding matched contract to proceed—must be distinguished from models of decentralized trade such as in Osborne and Rubinstein (1990). In their models, matches of buyers and sellers are random with the short side of the market certain of a match while all members of the long side have equal probability of a match. After pairings are made, the matched buyer and seller bargain to determine the terms of trade. The process might be repeated over time, or repeated until players drop out of the market by concluding an exchange. These models include a mechanism to determine matches, but the mechanism does not use players' strategies in deciding which matches to make.
Outside the principal-agent framework, our model is related to the non-cooperative assignment (or matching) problem. After principals and agents have made their announcements, our mechanism is used to solve an assignment problem, matching (some) principals with (some) agents in a manner consistent with the "preferences" expressed by their strategies. We have particular constraints on the way preferences may be expressed: principals announce contracts and agents announce sets of acceptable contracts. Our model differs from the typical matching problem in that agents' preferences over principals depend upon the menus of contracts offered by principals. Our problem is most similar to the assignment problem when we consider competitive (as opposed to Nash) equilibrium. Gretsky, Ostroy, and Zame (1992) examine competitive equilibrium in a nonatomic assignment model. In their matching problem, each of a continuum of potential sellers has a house and a reservation value for it. Each of a continuum of potential buyers has a reservation value for each house. When agents are risk neutral, our model can be interpreted in the context of their general assignment problem to show competitive equilibria exist.
As in both the decentralized trade and assignment literature, we do not explain where the mechanism assigning matches comes from. The properties we impose on the mechanism seem reasonable and desirable, but we do not exhibit a particular mechanism and say "this is the way the world works."
Within the principal-agent framework, related literature includes analysis of multiple agents (principals) competing to be matched with a single principal (agent). For example, in McAfee and McMillan (1987) n risk neutral agents compete for a contract with a single principal. In Biglaiser and Mezzetti (1992) two principals compete for a single risk neutral agent. Hermalin (1992) examines the interaction of a single principal and single agent when an exogenous parameter reflecting competitiveness changes. However, his competitiveness is in terms of the market for the principal-agent pair's output, not principals competing for agents or agents competing for principals.
The most closely related papers in terms of competition for principals and/or agents are Foster and Wan (1984, 1987) in which M identical, risk neutral firms and L risk neutral workers compete, with firms allowed to hire multiple workers. The equilibrium concept "is motivated by a tatonnement process with the expected utility of contracts playing the role of prices". As the number of firms increases relative to the number of workers, workers' equilibrium ex ante expected utility increases.
The paper is organized as follows. Section 2 introduces the framework of principals and agents as well as the nature of the game and the mechanism for matching principals and agents.
Section 3 examines an example of a matching mechanism. Section 4 contains results for general mechanisms when the contributions of principals and agents to their match (with a first-best contract) are separable. Section 5 contains some simple results for the non-separable case as well as for the case of risk averse agents.
2. Principals and Agents
In this section we introduce the framework for our model of a market for principals and agents. A continuum of principals will each offer a menu of contracts while a continuum of agents will each announce a set of acceptable contracts. Then an allocation mechanism will match (some) agents with (some) principals and contracts.
The set of potential agents, B, and the set of potential principals, S, are both compact subsets of the set of positive real numbers. Assume the mass of potential agents and the mass of the potential principals are nontrivial; i.e., p.(B) and p.(S) are nonzero where p, is Lebesgue measure. Each potential principal sES has a single project which can be completed only with the services of an agent. The outcome of the project is random, with the distribution depending upon the project, s, the (ability level of the) agent, b, and the effort level of the agent, e. Let x(s,b,e) be the random outcome.2 Each principal is risk neutral. The project cannot be transferred to any other principal but the project may be foregone, in which case principal s obtains the finite expected payoff v(s) by investing elsewhere. This outside opportunity for the principal is not usually explicit in principal-agent problems. If, as in many models, the principal must hire an agent, then implicitly v(s) - Sometimes, at least implicitly, models allow the principal to refuse to hire an agent if the expected payoff to the principal is negative; i.e., v(s) = 0.
Each potential agent bEB is risk neutral and has a utility function which is separable in payment and effort: W(t,e,b) = t - h(e,b) where t is payment received and e is effort level. We assume hj > 0, hn > 0 and h(0,b) = 0 where subscripts indicate partial derivatives. Each agent may work on any single project or outside this market. If the agent chooses not to work for a principal, by working in another market she obtains the finite expected utility U(b) > 0.
2 For notational simplicity we write x as a function of s and b directly rather than as a function of "project type" and "ability level". We do not assume higher s or b implies better projects or ability levels.
Information about each principal is common knowledge as is information about the distribution of agents' characteristics. Each agent knows her own characteristics but no principal can distinguish any two agents. When an agent works for a principal, the effort level chosen by the agent is not observable by the principal. The only jointly observable and verifiable
information upon which a contract may depend is the realization of the random outcome, x(s,b,e). Since different principals may have projects of different "types," in equilibrium
different principals may offer different contracts and still attract agents. A principal may offer a menu of contracts, but only a single contract employing a single agent may be completed.
The game among principals and agents proceeds as follows. Potential principals and potential agents simultaneously and independently make announcements. Each is attempting to maximize expected payoff. Each principal, s, announces a menu of contracts, i.e., an index set I(s) and for each i G I(s) a contract t(x,s,i). If principal s is matched with an agent in the allocation via contract i, and the realization of the random outcome for his project is x, then t(x,s,i) is the payment from s to his agent. Each potential agent announces a set of acceptable contracts. We restrict the possible strategies for each potential agent, b, by requiring that each agent choose a target utility level, U*(b), and accept any contract with which she could obtain expected utility at least U*(b). A contract with principal s paying t(x) to the agent when
outcome x arises is in the acceptance set of agent b if and only if there is an effort level, e, such that working for principal s at effort level e with contract t(x) leads to expected utility for b of at least U*(b). Let A(b) be the corresponding set of acceptable contracts. A(b) contains all
conceivable contracts that lead to expected utility U*(b) or higher. Thus if A(b) D A(b'), then b is willing to accept contracts b' would refuse.
Given any t(x,s,i), let K(b,s,i) be the largest number such that a contract paying t(x,s,i) - K(b,s,i) by principal s is acceptable to b. Then t(x,s,i) is acceptable to b if and only if K(b,s,i) >
0. We say agent b strictly prefers t(x,s,i) to t(x,s',i') if K(b,s,i) > K(b,s',i').
An allocation mechanism, to be described later, will be used to match agents with principals and contracts. After principals and agents have been matched, those principals
without a match invest elsewhere and obtain their outside alternative, v(s); those agents without a match obtain employment elsewhere and receive expected utility U(b); those principals and agents with a match proceed with a standard principal-agent relationship; i.e., the agent chooses an effort level, an outcome x is realized, and the contractual payment t(x,s,i) is made by the principle to the agent. The expected payoffs depend upon the matches made (since the outcome random variable, x, depends on project type, the agent's ability, and the agent's effort level). The structure of the game and the allocation mechanism are common knowledge.
We now consider the properties of the allocation mechanism. Given the contracts offered by the principals and the announcements of acceptable contracts by the agents, an
allocation mechanism matches principals and contracts with agents. If ((s,i),b) and ((s',i'),b') are two distinct matches made by the mechanism, then b # b' (an agent can have at most one
contract) and s # s' (a principal can have at most one contract in use). Not all principals and agents need be matched by the mechanism. The outcome for a matched principal and agent is determined by the match and the resulting effort choice and random outcome. We assume the allocation mechanism cannot force participation in unacceptable contracts. Thus if ((s,i),b) is a match made by the mechanism, then i is in the index set announced by principal s and t(x,s,i) is acceptable to agent b. The mechanism must be measure preserving in the sense that for any sets B 'c B and S 'c S, if the agents in B' are matched with the principals in S' then p.(B') = p(S'). We also impose the following properties on any allowed allocation mechanism. In Section 3 we study a specific example satisfying these properties; in Section 4 we look at general mechanisms which satisfy the three properties.
P l. At any allocation resulting from the mechanism, there do not exist an s E S, i E I(s) and b E B such that t(x,s,i) is acceptable to b but neither s nor b is matched.
This is an efficiency property. No obvious matches should be missed by the allocation mechanism.
P2. Consider any allocation resulting from the mechanism, when t(x,s,i) and t(x,s',i') are available contracts and A(b) and A(b') are sets of acceptable contracts for b and b', respectively.
a. If all agents strictly prefer t(x,s,i) to t(x,s',i'), and t(x,s',i') is matched, then t(x,s,i) is matched.
b. If A(b') is a strict subset of A(b), and b' is matched, then b is matched.
c. If A(b') is a strict subset of A(b), for any contract in A(b'), b is at least as likely as b' to be matched with that contract.
We allow the mechanism to employ random selections with the provision that it is not biased against any principal or agent, and is positively responsive to "accommodating behaviors." By (a), by offering a contract all agents prefer, a principal is more likely to be matched. By (b), by being willing to accept more types of contracts, an agent is more likely to be matched. By (c), if b is willing to accept contracts acceptable to b', as well as others unacceptable to b' (i.e., ones that are too bad for b'), b should not necessarily be matched with contracts unacceptable to b' but instead should have an equal chance of being matched with the better contracts. This property will be used as follows. If a group of principals with strictly positive measure is matched with a group of agents using contracts which are very beneficial to the agents, if an additional agent
"undercuts" these agents by offering a larger acceptance set, then the undercutter will obtain one of these desirable contracts with probability one (since it is "undercutting" a positive mass of agents).
P3. If the set of contract offerings and announcements of acceptable contracts changes for only a set of principals and agents of measure zero, then the matches made by the mechanism change for only a set of principals and agents of measure zero.
This is a "continuity" assumption on the mechanism. A negligible change in the profile of strategies leads to a negligible change in the resulting allocation.
In the standard model with one risk-neutral principal and one risk-neutral agent of known type, the optimal contract for the principal is to "sell the firm" to the agent by setting t(x,s) so that x-t(x,s) is constant (i.e., the principal receives a fixed payment independent of the outcome) and the agent, using the "first-best" effort level, just obtains her outside alternative utility level.
The agent is induced to work at the "first-best" effort level since she receives all the benefits of extra effort. In our model, ignoring the alternatives outside the market, the maximum potential gross "gains from trade" in a match of principal s and agent b are T(s,b) = Ex(s,b,e*(s,b)) - h(e*(s,b),b) where E is the expectation operator and e*(s,b) is the effort level that maximizes Ex(s,b,e) - h(e,b). Taking account of the alternatives outside this market, the maximum potential net "gains from trade" in a match of principal s and agent b are T(s,b) - U(b) - v(s). In the
standard model with a single principal and a single agent of known type, the principal sets the
"sell the firm" contract t(s,x) - x + U(b) - T(s,b), the agent accepts the contract and works at effort level e*(s,b) to obtain expected utility U(b), and the principal obtains T(s,b) - U(b). The function T(s,b) will play an important role in our analysis.
We will show the Nash equilibria for our game correspond to competitive equilibria;
hence, we must define a competitive equilibrium for the market. Since different principals have different projects, they are selling different "goods." Thus the "price" may differ. Our definition of competitive equilibrium will be presented in a slightly unusual way to emphasize the nature of the rents obtained by the different principals and agents.
Definition: A competitive equilibrium is a base price, K, and rents, R(s), to principals,3 and matches between (some) principals and (some) agents such that:
3 There is an indeterminacy in K and R(s) since they always appear as K + R(s). However in some applications it is useful to distinguish the two by setting the minimum of the R(s) values among matched principals at zero and adjusting K accordingly. Note the principal receives K + R(s) only if matched.
(i) principal s is matched (and receives the fixed amount K + R(s) ) if v(s) < K + R(s) and only if v(s) < K + R(s);
(ii) agent b is matched if T(s,b) - U(b) > K + R(s) for some s and only if T(s,b) - U(b) > K + R(s) for some s. A matched agent b is paired with a principal s who yields the highest gain (i.e., with an s* G argmaxs{T(s,b) - U(b) - K - R(s)} ) and receives expected utility T(s*,b) - K - R(s*).
(iii) The matching is measure preserving in the sense that for any sets B 'c B and S 'c S, if the agents in B' are matched with the principals in S' then p(B') = p(S').
By Theorem 4 of Gretsky, Ostroy, and Zame (1992), a competitive equilibrium exists if T(s,b) - U(b) - v(s) is an upper semi-continuous function,4 which we assume.
In the next two sections we consider the case in which the "gross gains from trade" are additively separable: T(s,b) = Ti(b) + T2(s). In this case there is an unambiguous ranking of projects and all agents agree on the extra value attached to working on project s rather than s'. In equilibrium, principals with better projects (i.e., higher T2(s) ) will earn higher rents, with the base level of rent determined by the marginal project matched with an agent. The base rent will be zero if there are unmatched principals with projects almost as good as that of the m arg in al
principal. If there are no such unmatched principals, the base rent could be strictly positive. In any competitive equilibrium with a positive measure of matched principals, if s and s' are both matched then the difference in rents, R(s) - R(s'), must equal the difference in "productivities"
T2(s) - T2(s') in order to prevent all agents from preferring the principal with the higher T2 - R value. The sets of principals and agents who are matched may matter, but the specific pairing between principals and agents from these sets does not. Either all principals are identical or any differences are exactly counterbalanced by the differences in rents they receive in equilibrium.
4 The function f is upper semi-continuous at x if f(x) is at least as large as the limit of f(xn) for any sequence of points xn converging to x.
Section 3 considers an example of an allocation mechanism and Section 4 considers general results for the separable case. Section 5 examines the nonseparable case where the actual pairing of principals and agents matters.
3. An Example of an Allocation Procedure
In this section we examine a specific allocation mechanism. It is just an example of a mechanism satisfying properties P l, P2 and P3. To facilitate understanding of the mechanism, we introduce the ideas in the context of a market for a single homogeneous good, with neither principals nor agents, but sellers taking the role of principals and buyers taking the role of agents. This will make it easier to understand the added complications necessary when we turn to the principal-agent model. The market has a continuum of both buyers and sellers, with each seller, s G S, initially holding one unit of a good which he is willing to sell at any price above his reservation value v(s). Each buyer, b G B, is willing to buy one unit at any price below her reservation value r(b). Assume r and v are measurable functions. The usual perfectly
competitive equilibrium price for this market is such that the "number" of buyers and sellers is equal at that price, i.e., p* such that p ({s G S | v(s) < p*}) = p ({b G B | r(b) > p*}).5
The game proceeds as follows. Potential sellers and buyers simultaneously and independently make announcements. Each seller announces a selling price, p(s) for seller s.
Each potential buyer announces a maximum price she is willing to pay, q(b) for buyer b. To determine the payoff to each player we must determine the allocation of the good among buyers and sellers and the prices at which exchanges occur. The mechanism proceeds through two steps. The first step determines which potential buyers will be allowed to buy and which potential sellers will be allowed to sell. We will say claims to buy are acquired and rights to sell
5 If reservation values are not dispersed, or there are gaps in the distribution of reservation values, or p(B) = p(S), p* is an equilibrium price if p({s G S | v(s) < p}) - p ({b G B | r(b) > p}) is nonnegative for p > p* and nonpositive for p < p*.
are established at different prices, but no exchanges are occurring and no money is changing hands in this step. After the identities of buyers are sellers are established, in the second step we make matches and exchange money and the good. Some technical details of the procedure for step one will be discussed after we complete the description of the two step process.
First, consider the determination of which buyers will be allowed to buy and which sellers will be allowed to sell. Until we get to the end of step one, each seller we discuss will get the absolute right to sell his unit. At the final price discussed in step one, if a mass of sellers have all announced that price, we may need to split them into those allowed to sell and those not allowed to sell. On the buying side, we are only determining potential claims to buy. Some buyers may drop out of the market in the first step and forfeit all claims to buy. Their claims will be reallocated to other potential buyers. No sales are taking place during this first part of the procedure. We are determining the identities of buyers and sellers only.
For any price p, all buyers with q(b) > p have equal claim to all units sellers have offered to sell at price p. Since there may be more claims to buy at price p than there are offers to sell, we assume each buyer gets the same fraction of a claim. The fraction, o(p), is equal to the number of units offered by sellers at price p divided by the number of buyers with claims. The procedure starts at the lowest selling price offered by any potential sellers, allocating claims to buyers and rights to sell to sellers. This is repeated at successively higher prices. Each buyer collects a fraction of a claim at each price up to the price, p, where either the buyer has already obtained a total of one claim for the good, or p = q(b). Note that if buyer b has obtained a total of one claim at prices less than or equal to p then any buyer b' with q(b') > q(b) must also have obtained a total of one claim since both buyers have equal claim on all units at prices less than or equal to q(b). Thus all buyers who receive a total of one claim in the allocation receive their final fraction of a claim at the same price.
If some buyers have q(b) = p but have not obtained a full claim, then they are dropped from the market and their partial claims are reallocated equally to remaining buyers. Thus it is as if these dropouts never existed in the preceding description of the mechanism. If a strictly
positive mass of buyers have q(b) = p but have not obtained a full unit, those with the highest b are dropped from the market, and their claims are reallocated until either all remaining buyers have a full claim, or all buyers with q(b) = p have been removed.
The procedure continues at successively higher prices, p, until either the buyers with claims at price p have each already obtained a complete claim, or p exceeds the highest announced selling and buying price.
In the second step, given the identities of the actual buyers and sellers as established in step one, buyers are matched with sellers and money and the good are exchanged. The actual sellers are ranked by announced selling price (and name, s, if there are ties), from lowest to highest. The actual buyers are ranked by announced maximum buying price (and name, b, if there are ties) from highest to lowest. The matches are made between lowest price sellers and highest price buyers, and this continues in a measure preserving way until all actual sellers and actual buyers are matched.
Now consider the allocation of money between buyers and sellers. The mechanism uses the seller's announced price when an exchange occurs. A seller always either retains his unit of the good or gives up his unit in exchange for the price he announced. Buyers pay their matched seller's price, p(s), when they purchase the good.
Given the allocation of the good and the prices paid as described above, payoffs are as follows. If a buyer does not obtain a unit, her payoff is zero. For a buyer, who receives a unit in a match with s, the payoff is r(b) - p(s). The payoff to the seller is p(s) - v(s) if the unit is sold and zero otherwise.6,7
We now turn to the technical details of the allocation of claims in the first step. The difficulty is that we must deal with both continuous and discrete distributions simultaneously.
6 if p** is the supremum of prices at or above which a positive mass of sales occur and p ({s £ S | p(s) = p**}) > 0, it may be necessary to ration sales among the firms with p(s) = p**, by letting the firms with lowest s sell. Note the firms need not be indifferent about selling since v(s) < p(s) is possible.
Evaluation of payoffs requires measurability properties of the strategy profiles, p(-) and p(b). These are necessary to evaluate the fraction of a claim each buyer gets at each price.
All potential sellers might announce different prices, or some or all might announce the same price. To deal with this we need to break up the distribution into two parts, corresponding to the
"smooth" and "lumpy" portions. This is most conveniently done in terms of the distribution of announced prices. Thus we first consider the induced distribution of prices, divide it into
"lumpy" and "smooth" portions, and finally determine the fraction of a claim obtained by a buyer at any price p.
If p ( ) is a Lebesgue measurable function, define the induced measure u on prices as follows: for any Lebesgue-measurable set E of prices, u(E) = p({s G S | p(s) G E}). The induced measure u specifies the "number" of sellers announcing a price p G E. (For example, the competitive supply at price p corresponding to the price announcements p(-) is u(f(),p]).) The measure u can be decomposed into two parts, u = tjq + u 1, where u () is singular with respect to Lebesgue measure and tj-j is absolutely continuous with respect to Lebesgue measure. The singular part, u 0, assigns positive measure only to the countable set of prices, p', at which a positive mass of sellers all agree to sell: v 0({p'}) = p. ({s G S | p(s) = p'}) > 0. This takes care of the "lumps" in the distribution of announced selling prices. The absolutely continuous part,
can be represented in terms of Lebesgue measure via a "density function" f, which is nonnegative and Lebesgue-measurable: for all Lebesgue-measurable sets E of prices, u-^E) = J fdu,. This takes care of the "smooth" part of the distribution of announced prices. The total
E
"number" of sellers announcing a price in set E is u(E) = 'Vq(E) + ^ ( E ) .
Let p** be the highest price at which units are actually sold in a particular allocation (i.e.
p** is the supremum of the set of prices at or above which a positive mass of sales occur). Let M be the smaller of p({s G S | p(s) < p**}) and p({b G B | q(b) > p**}). M is the total mass of units which will be exchanged. Buyers with q(b) > p** will obtain a unit. Buyers with q(b) <
p** wiH not obtain a unit. Mass M - p({b G B | q(b) > p**}) of buyers with q(b) = p** will obtain a unit. In the following description, active buyer means a buyer who obtains a unit. For p < p**, if 'u0({p}) > 0 (i.e., a positive mass of sellers announce p), then the fraction of a claim each active buyer obtains is o (p) = u 0({p})/M. It is the ratio of the mass of units for sale at
price p and the mass of active buyers with announced willingness to buy at price p. Note this fraction is non-infinitesimal. At p - p**, if u 0({p**}) > 0 (i.e., a positive mass of sellers announce p**), and the mass of active buyers with claims at p** is also positive, it may be necessary to ration sellers. The fraction o(p**) will be less than the formula given above for o(p) if buyers would obtain a total of more than one unit. In that case, o(p**) is the value that leads to a total of exactly one unit for the active buyers, and some sellers with p(s) = p** do not sell the good. For p < p**, if "^({p}) = 0, then o(p) = f(p)/M where f is the density of u, relative to q. This fractional share is infinitesimal since the mass of sellers at p is zero. The total fraction of a claim obtained by active buyer b is J o odp, + where E* - {p | TJo({p}) > 0} (T [0, p**]. The first term is the "smooth" integral over the firms who offer a dispersed set of prices while the second term is the "lumpy" sum over the prices at which masses of firms concurred in their price announcements. By the properties of the mechanism, each active buyer obtains one unit. The price paid by active buyer b matched with seller s is p(s).
At this point an example to demonstrate the method of determining payoffs may be useful.
Example: Assume the mass of potential sellers is two, S = [0,2], and the mass of potential buyers is 1.5, B = [0,1.5]. Suppose the announced strategies are p(s) = s for s G [0,1], p(s) = 0.5 for s G (1,2], and q(b) = 1.5 - b for all b G B. The sellers form two groups. For s G [0,1], sellers' prices are dispersed uniformly between zero and one. For s G (1,2], all sellers set the same price, p = 0.5. The corresponding induced measure u = i>0 + is as follows. The only
"lump" is at p = 0.5 with mass one of sellers so TJ0({p}) = 0 for all p * 0.5 and tjq({0.5}) = 1.
Mass one of sellers have prices uniformly distributed over [0,1] so the appropriate density is f(p)
= 1 for p G [0,1] and zero otherwise. Since q({s G S | p(s) < p}) - q({b G B | q(b) > p}) changes sign at 0.5, p** = 0.5. Thus M = q({b G B [ q(b) > 0.5}) = 1 and all b G [0,1] are active buyers.
For p < 0.5, o(p) = f(p)/M = 1 is infinitesimal. For p - 0.5, a(p) = u 0({0.5})/M = 1 is non
infinitesimal. Sellers must be rationed at p**. Buyers b G (1,1.5] have q(b) = 1 - b < 0.5 so
they do not obtain a claim and pay nothing to the sellers, for a payoff of zero. Buyers b £ [0,1]
obtain 0.5 of a claim at prices below 0.5 and the remaining 0.5 of a claim at price 0.5. Sellers s
£ [0, 0.5] sell their units at p(s) = s for payoff s - v(s), and sellers s £ (0.5, 1] do not sell, for payoff 0. In aggregate, 0.5 unit is sold at price 0.5 so the sellers s £ (1,1.5] sell their units at p(s) = 0.5 for payoff 0.5 - v(s), and the remaining sellers s £ (1.5,2] do not sell for payoff 0.
Buyer b £ [0,0.5] is matched with seller s=b for payoff r(b)-s. Buyer b £ (0.5,1] is matched with seller s = b + 0.5 for payoff r(b) - 0.5.
A Nash equilibrium for the game is a pair of measurable functions, (p(-), q(-)), the strategy profiles, such that each principal and agent is maximizing expected utility given the strategies of the others. It is easy to see that any (price-taking) competitive equilibrium price, p*, and corresponding competitive equilibrium allocation of expected utilities can be supported as a Nash equilibrium outcome.8 Let all potential sellers who sell in the competitive equilibrium set p(s) = p* while other potential sellers set p(s) = v(s).9 Let all potential buyers set q(b) = r(b).
This is a Nash equilibrium in which all units sold sell at the competitive equilibrium price, p*, and the Nash equilibrium allocation of utilities is also a competitive equilibrium allocation of utilities for price p*.
As in other games of this sort, there may be a no-trade Nash equilibrium in which all sellers announce higher values for p(s) than buyers announce for q(b). For example, if r = sup{r(b) | b £ B} < oo, then p(s) = r and q(b) = 0 is an equilibrium. However, as we show below, the outcome in any Nash equilibrium with trade corresponds to a competitive equilibrium. The result follows from the fact that with a continuum of potential buyers and sellers, each buyer views the fraction of a claim that will be obtained at each price, o(p), as
When some buyers or sellers are indifferent about being in the market at p*, by
assumption our mechanism generates the allocation of the good in which the low b buyers or low s sellers whose strategies reflect indifference actually participate. However, the allocation of utilities is the same in the Nash equilibrium as in the competitive equilibrium.
9 If p.({s £ S | v(s) = p**}) > 0, then this may need to be modified to ensure that sellers with v(s) < p* get to sell. The problem is our mechanism uses "name" as tie breaker.
independent of her own strategy, and each seller views the highest price paid for a unit of the good that is sold as independent of his own strategy. Thus each seller either offers to sell at this highest price or does not want to sell. Each buyer with reservation value above this price wants a unit while each buyer with reservation value below this price does not want to buy. The allocation preserves the total number of units, so roughly speaking, supply equals demand at this price. The resulting allocation of expected utilities coincides with allocation of expected utilities in a competitive equilibrium.
Theorem 1: Let (p(-), q(-)) be a Nash equilibrium at which the measure of units traded is strictly positive. Then all transactions take place at the same price, p**. The price, p**, is a competitive equilibrium price. The Nash equilibrium allocation of utilities is a competitive allocation of utilities at price p**.
Proof: Let (p(-), q(-)) be a Nash equilibrium at which the measure of units traded is strictly positive and let p** be the supremum of prices at which units are actually sold. To be
optimizing, all sellers with reservation values v(s) less than p** must be selling their units, and no announced price p(s) < p** could be optimal for them. All sellers with reservation values greater than p** must not be selling, and no announced price p(s) < p** could be optimal for them. Sellers with v(s) = p** may or may not be selling, but p(s) > p** for them. Thus p(s) >
p** for all s and the measure of units sold lies in the range [p ({s £ S | v(s) < p**}), p ({s £ S j v(s)<p**})].
By the nature of the procedure for allocation of units to buyers, if q(b) < p** then b could not have obtained a unit. However, q(b) > p** must lead to allocation of a unit since some buyers are purchasing the good. Thus all buyers with reservation values greater than p** must be purchasing a unit, and q(b) > p** for them. Since all sellers set p(s) > p**, all buyers with reservation values less than p** must not be purchasing at all; i.e., either q(b) < p** or q(b) =
p** and buyers (including these) are rationed at p**. Buyers with r(b) = p** may or may not be purchasing. Thus the measure of units purchased lies in the range
[p ({b e B | r(b) > p**}), p ({b £ B | r(b) > p**})].
At the Nash equilibrium allocation the measure of units purchased equals the measure of units sold, so the intervals of possible sales and purchase levels must overlap, or
[p ({b e B | r(b) > p**}), p ({b 6 B | r(b) > p**})] D [p ({s e S | v(s) < p**}), p ({s G S | v(s)
# 0 . But this implies p** is a competitive equilibrium price for the price-taking model of this market. Note all transactions take place at price p**, and the Nash equilibrium allocation of goods and utilities is also a competitive equilibrium allocation of goods and utilities for price
Now consider a market for principals and agents in which T(s,b) is additively separable and all principals are constrained to offer only "sell the firm" contracts; i.e., for all s and i, t(x,s,i) is such that x - t(x,s,i) is a constant.10 Clearly there is no role for menus of contracts since all agents would agree on the ranking of contracts offered by any s. With this restriction, the strategies are very similar to those of the homogenous good example. Each principal sets a price for "selling his firm" and each agent has acceptable prices for each firm. The added complication is that principals are not necessarily selling the same commodity since the distribution of the random outcome depends not only on the agent's ability and effort level but also on the specific project. Thus for a given U*(b), agent b may be willing to pay different amounts to different principals based on the different "productivities" of their projects. We
10 Without this assumption the allocation procedure matching principals and agents becomes substantially more complicated. Unfortunately we cannot argue "in equilibrium only 'sell the firm' contracts are offered," or "if other principals are offering 'sell the firm' contracts it would be in the best interest of principal s to also be offering such a contract" without first specifying the allocation procedure with general contracts, something we do not wish to do in this example. In the next section we treat more general mechanisms which allow general contracts.
assume T(s,b), U(b), and v(s) are measurable functions and T(s,b) - U(b) - v(s) is an upper semi- continuous function.
The allocation procedure for the homogeneous commodity game must be modified to account for agents' willingness to pay different amounts to work for different principals. Given the restriction on agents' strategies (they choose target utility levels U*(b) and accept all
contracts leading to at least the target utility level), for any strategy profile all agents have the same differential in willingness to pay for s versus s', T2(s) - T2(s'). The allocation mechanism will proceed through two steps: first, deciding which principals will be matched and which agents will be matched, and second, assigning the actual matches. The two steps will proceed as in the homogenous good case with some minor alterations.
Consider the first step as in the homogenous good case. Rather than starting with the lowest prices offered by the principals, the procedure starts with the lowest net price taking account of the differential in willingness to pay for s versus s'. The process proceeds as follows.
Pick a "standard commodity", say the project of principal s'. The process will use the price of this standard project, z, as the analog of the price in the process for the homogenous good. The adjusted price for s is z + T2(s) - T2(s'), and all agents are indifferent between s' at price z and s at price z + T2(s) - T2(s'). Agents play the role of buyers. For each level of z, an agent is
"willing to buy" if the "standard commodity" (i.e., the project available from s') is acceptable at
"price" z (i.e., the contract paying s' the fixed amount z). Principals play the role of sellers. For each level of z, principal s is "willing to sell" if z + T2(s) - T2(s') is at least as large as the fixed payment to s in the contract offered by s. With these modifications the first step of the
mechanism proceeds exactly as in the homogenous good case. At each "price" z, the set of principals "willing to sell" and agents "willing to buy" is used to determine the absolute right of principals to "sell" and agents' "claims to buy." As in the homogenous good case, the outcome of this step is a set of principals who will be matched, and a set of agents who will be matched.
In the second step specific matches are made. This step proceeds as in the homogeneous good case except the principals are first ranked not by "prices" but by "adjusted prices," where
the adjustment takes into account their productivity premium T2(s) - T2(s'). Each principal who is matched obtains the fixed payment called for in his "sell the firm" contract. An agent who is matched with principal s obtains the corresponding project (with random outcome) for the fixed payment called for in the contract.
We are now able to apply the results of Theorem 1 to obtain analogous results for the principal-agent market.
Theorem 2: Let T(s,b) = Ti(b) + T2(s). Any competitive equilibrium base price, K, rents to principals, R(s), and matching of principals and agents can be supported as a Nash
equilibrium.11 In any Nash equilibrium with a positive measure of matches, all matched
principals use contract t(x,s) = x - K + T2(s)12 where K is a competitive equilibrium base price, T2(s) yields the corresponding competitive equilibrium rents to principals, and the Nash
equilibrium matching is a competitive equilibrium matching for base price K and rents R(s) s T2(s).
Example: Suppose U(b) = 0 and h(e,b) = e2 for all b G B, and v(s) = 0 for all s G S. Since principals and agents are risk neutral, we specify the expected outcome given s, b, and e directly without specifying the distribution: Ex(s,b,e) = 2e(b+s)1^2. Then the optimal effort level is e*(s,b) = (b+s)1/2 and the potential "gains from trade" are T(s,b) = b+s. The equilibrium depends on the sets S and B. If B = [0,1] and S = [0,4] then in equilibrium s* = 3 is the
marginal principal, receiving K = 0. Principals s > 3 are matched and receive the base price K = 0 plus a rent equal to their additional "productivity" beyond s* = 3, for a total payment of s-3.
Principals s < 3 are not matched, and receive 0. All agents are matched, and have expected
H As in footnote 8, when some principals or agents are indifferent about being in the market, by assumption the mechanism generates only the allocation in which the low b or low s individuals who are indifferent actually participate.
12 The contract must in effect be of this form, though its actual structure may differ. The crucial point is that the contract must lead to a first-best effort level and expected utilities for each principal and agent must match those of a competitive equilibrium.
payoff b + 3 for b e B. If on the other hand B = [0,4] and S = [0,1], then in equilibrium b* = 3 is the marginal agent. All principals are matched and the "least productive" principal, s = 0, receives base price K = 3. Principals s > 0 receive the base price plus a rent equal to their
additional "productivity" beyond that of the worst principal, for a total payment of s + 3. Agents b < 3 are not matched and receive 0. Agents b > 3 are matched and have expected payoff b - 3.
Note the marginal player (s* = 3 in the first case, b*= 3 in the second) receives 0 (the level of opportunity outside this market) and other players obtain strictly positive expected payoffs.
4. General Mechanisms in the Separable Case
We now turn to more general mechanisms and general contracts in the separable case.
As in the previous section, it is easy to see that a competitive equilibrium allocation can be supported as a Nash equilibrium. If (1) the principals who were matched in the competitive equilibrium offer their competitive equilibrium "sell the firm" contracts, (2) the agents who were matched in the competitive equilibrium set target utilities equal to their competitive equilibrium expected utility levels, and (3) the principals and agents who were not matched in the
competitive equilibrium are assigned strategies which leave them unmatched in the game, the strategies form a Nash equilibrium and lead to the competitive equilibrium allocation of expected utilities.
We are interested in the question of whether the Nash equilibrium outcome always leads to competitive equilibrium allocations of expected utility. If that is the case, then all Nash equilibrium contracts that are matched with agents are essentially "sell the firm" contracts in the sense that they lead to first-best effort choices. Also, there would be no role for menus of contracts to sort agents since the relevant reservation utility for each matched agent is determined endogenously in the equilibrium and coincides with the competitive equilibrium level.
We will examine this question in two different contexts. The first starts from the previous section and generalizes the mechanism (to mechanisms satisfying Pl, P2 and P3) and allowed contracts (to any t(x,s,i) ) while adding a restriction which limits the ways in which agents may differ from each other. The proof that all Nash equilibria with a positive mass of matches correspond to competitive equilibria follows the ideas of the proof in the previous section. This is because the restriction on agent variability will maintain the problem as one dimensional even with general contracts. (The separability condition by itself is not enough to do this. It is based on first-best effort levels, which may not be attained with arbitrary contracts.
Then different agents may have different rankings of the principals' offers.)
In the second context, the restriction limiting ways in which agents differ is dropped.
Allowing general mechanisms, contracts, and variability among agents, we focus on Nash equilibria in which unmatched principals and agents "tell the truth" in the sense that their strategies are the most attractive (to the other side of the market) contract they can offer without risking being worse off with a match. For agents, this means the target utility level is their reservation utility from outside the market. For principals, this means they offer the "sell the firm" contract with "price" equal to their outside opportunity. In this context, any Nash
equilibrium with "truth telling" by unmatched players corresponds to a competitive equilibrium.
First consider restrictions on the ways in which agents may vary. The restriction will allow us to get a consistent "ranking" of agents which is independent of the contracts. There are several possible restrictions that produce consistent rankings. If agents are identical except for their reservation utility outside the market, U(b), then we obviously will get consistent rankings.
If agents are identical except for their disutility of effort functions, h(e,b), then with appropriate assumptions on the variation in h, a consistent ranking can be produced. We will consider a third possibility which is related to the second. All agents are identical except for a productivity parameter, a, contained in a compact subset of (0,°°). The distribution of the final outcome depends on the project, s, and the level of effective effort, F(e,ö), where F is differentiable with
strictly positive first partial derivatives. We assume all first-best effort levels for all agents are contained in a compact set.
A l. The set of potential ability levels, D, is a compact interval. The function g:B—>D that specifies the ability level of each agent is continuous. The function U(b) is also continuous.
If b is more productive in net gains from trade than b' (i.e., Ti(b) - U(b) > Ti(b') - U(b') ), then for any acceptance set A(b') in which b' is no worse off than U(b'), b can choose an acceptance set A(b) containing A(b') and be no worse off than U(b). This allows us to use property P2 of the general mechanism. Since agent b is the more productive agent, agent b can always
"undercut" agent b' if it is desirable. On the other hand, condition A l guarantees that if b is earning a rent in some allocation, then there is some b' who is slightly less productive than b who could undercut that rent level. If will be in the interest of b' to do this unless b' is also earning an appropriate rent. Thus an agent will not be able to earn too high a rent unless her slightly less productive fellow agents are earning almost as much rent.
Theorem 3: Assume T(s,b) = Ti(b) + T2(s) and the allocation mechanism satisfies Pl, P2 and P3. If the set of agents satisfies A l, then in any Nash equilibrium with a positive measure of matches, the allocation of expected utilities coincides with a competitive equilibrium allocation of expected utilities.
Proof: Consider a Nash equilibrium with a positive measure of matches. If s' is matched, and T2(s) - v(s) > T2(s') - v(s'), then by P2, s can "undercut" s' by offering a contract all agents find more attractive than that offered by s'. Thus the set of matched principals must be those with highest T2(s) - v(s) values.
If s has expected utility, m, then any matched s' has expected utility less than or equal to T2(s') - T2(s) + m (or s could "undercut" s' and do better). Thus if s and s' are both matched,
then the difference in expected utilities for s and s' must equal T2(s') - T2<s). This implies all matched contracts must induce first-best effort levels (or another principal could "undercut"
using a "sell the firm" contract and still be better off).
All matched agents must be at least as well off as with a "sell the firm" contract at "price"
v(s) from the unmatched principal with highest T2(s) - v(s) (or this principal could "undercut" a matched principal).13
If b' is matched and Ti(b) - U(b) > Ti(b') - U(b'), then by A l and P2, b can "undercut" b' by offering an acceptance set strictly containing A(b'). Thus the set of matched agents must be those with highest Ti(b) - U(b) values. If the matched agents of lowest ability level are earning rents that exceed U(b) by an amount that is bounded away from zero, then by A1 and P2c the unmatched agent of highest ability level could profitably "undercut" these agents. By A l, for a little extra disutility of effort the unmatched agent could produce the same effective effort as the slightly higher ability matched agents, and by P2c, by offering a slightly larger A(b) than the slightly higher ability matched agents, the unmatched agent could get a favorable match.
Similarly, the expected utilities of matched agents must differ by their differences in Ti(b) - U(b), or they could "undercut" each other and be better off.
We have shown all matched contracts induce first-best effort levels, so the highest
possible net gains from trade are available to be distributed. The "marginal" unmatched principal determines a lower bound for agents' expected utilities. The "marginal" unmatched agent
determines a lower bound for principals' expected utilities. Expected utilities among matched principals (agents) differ by their differences in productivity, T2(s) - T2(s') ( Ti(b) - Ti(b') ).
The "marginal" matched principal and agent must exhaust all the net gains from trade in their matches, but must prefer being matched to not being matched. The only allocations satisfying
13 No unmatched s may attain the "highest" value because the set may not be closed. The argument then applies to all principals near the supremum of values. A similar note applies to agents in the following steps. For brevity we will speak of a highest value even if it is not attained.
all these properties are competitive allocations. Thus the allocation of expected utilities must coincide with a competitive equilibrium allocation of expected utilities.
Now consider general mechanisms, contracts, and variability among agents. We say an unmatched principal's (agent's) strategy was truthful if it was the most attractive strategy to agents (principals) it could offer without fear of being worse off than being unmatched. For principal s, the truthful strategy is to "sell the firm" at price v(s). For agent b, the truthful strategy is to use target utility level U(b) to determine A(b)
A matched principal's strategy is always truthful in the sense that he gets to use one of his offered contracts. We might say a matched agent's strategy is truthful if her target utility level is her equilibrium expected utility. Any competitive equilibrium allocation of expected utilities can be supported as a Nash equilibrium allocation of expected utilities using truthful strategies for all principals and agents.14.
For any Nash equilibrium with unmatched principals or agents, there is a class of
"equivalent" corresponding Nash equilibria in which matched principals' and agents' strategies are unchanged but unmatched principals' and agents' strategies vary in ways that maintain their unmatched status. Does this class include a Nash equilibrium in which unmatched principals and agents are truthful? If not, then the Nash equilibrium allocation can only be obtained when some principal or agents use strategies that demand more utility than they obtain in the
equilibrium, and thus remain unmatched. Such an equilibrium is not robust to reasonable changes in the strategies of unmatched principals and agents.
14 This statement need not be true if there is a positive mass of principals who are indifferent about being matched in the competitive equilibrium. We may need to distinguish those principals who strictly prefer to be matched so the allocation mechanism can make the proper matches. This could be done by allowing principals to indicate the contracts they have offered which they are indifferent about having matched. Then when faced with "ties" among principals to be matched, the mechanism would be required to match principals who did not acknowledge indifference before those who did. With this tie breaking rule, all competitive equilibrium allocations of expected utility could be supported as Nash equilibrium allocations using truthful strategies.
Theorem 4: Assume T(s,b) = Ti(b) + T2(s) and the allocation mechanism satisfies Pl, P2 and P3. In any Nash equilibrium in which unmatched principals and agents use truthful strategies, the allocation of expected utilities coincides with a competitive equilibrium allocation of expected utilities.
Note that with truthful strategies for unmatched principals and agents, it is not necessary to assume a positive measure of matches in the Nash equilibrium. If the measure of matches is zero, then it is zero in the competitive equilibrium. Truthful strategies for unmatched players removes the possibility of being stuck in a "no-trade" equilibrium when "trade" should occur.
Proof: As in the proof of Theorem 3, the set of matched principals must be those with highest T2(s) - v(s), differences in expected utilities among matched principals must correspond to differences in T2 values, and matched contracts must induce first-best effort levels. (This is true even if only a single principal is matched since the realized distribution of agents is known. The best agent type is available.)
Truthful unmatched principals and agents provide lower bounds on the expected utilities of matched agents and principals and guarantee that the measure of matches in the Nash
equilibrium must be a competitive equilibrium measure of matches. (Too many matches would mean payoffs worse than being unmatched for some players, while too few matches would violate Pl, given the truthful strategies of unmatched players.)
If the measure of matches is zero, it is zero in the competitive equilibrium, and it is easy to see the allocation of expected utilities coincides with a competitive allocation of expected utilities, for matched and unmatched players.
If the measure of matches is positive, the set of matched principals and the set of matched agents must coincide with corresponding sets in a competitive equilibrium. (The measures are equal, and in both equilibria the more productive types must be in the matched
sets). All matched contracts induce first-best effort levels, and there is a K such that each matched principal has expected utility T2(s) + K. By separability, matched agents must receive the appropriate expected utility level, and K must be such that neither principals nor agents are worse off than being unmatched (i.e., it must correspond to a competitive equilibrium level).
The only allocation of expected utilities consistent with these properties is a competitive equilibrium allocation.
5. Remarks on the Non-separable and Risk Averse Cases
Finally consider the case in which T(s,b) is not additively separable. Here pairings of principals and agents matter since even in equilibrium, a matched agent need not be indifferent among all the matched principals. In contrast to the previous cases, even with "sell the firm"
contracts, given a strategy profile, the set of principals whose contracts b finds acceptable need not have any (set inclusion) relationship to the corresponding set b' finds acceptable. Any allocation procedure for this game would need to deal explicitly with the possible infinity of types of principals. The actual pairings of principals and agents would need to be made carefully. Any such procedure would be much more complicated than that of Section 3. In Section 3, every active buyer had equal claim to each low priced seller, a simple, plausible allocation mechanism. In contrast, any allocation mechanism for the general T(s,b) case would require elaborate rules, procedures, and computations to keep track of the infinity of buyers and the infinity of commodities.
We can still discuss competitive equilibrium for the general T(s,b) case. Competitive equilibria exist and have expected properties in terms of rents and marginal principals and/or agents, but the equilibrium pairings may have interesting properties.
