The economy consists of a mass M of small, identical rms (principal, he) and a mass of N > M identical employees (agent, she). Principals and agents are risk neutral. The time horizon is innite, time is discrete (periods are denoted t = 1,2, ...), and all players share a common discount factor δ. As principals and agents are identical, they are not further indexed.
At the beginning of each period t, we distinguish between whether a player currently is part of a match or not (in t= 1, everyone is obviously unmatched). Each unmatched
principals can oer a contract to exactly one unmatched agent (we assume that this process is completely arbitrary and contains no frictions).5 This oer consists of a legally
enforceable wage payment wt which is not restricted to positive values. A principal who
does not make an oer gets an outside utility π in the respective period, where we make the normalization π = 0. If an unmatched agent receives no oer from any principal or
receives one and rejects it, she has to consume her exogenuous outside utility, which is normalized to zero as well. If she gets an oer and accepts it, she consumes wt and then
chooses an eort levelet∈[0,1]. This leads to outputyt=θwith probababilityet, and to
yt= 0 with probability (1−et). While the output is directly consumed by the principal,
the agent suers eort costs c(et), with c0, c00, c000 >0, c(0), c(0)0 = 0and c(1)suciently
large to never be optimal. After output realization, each agent - no matter whether employed or not - leaves the market for exogenuous reasons for example because the partner found a job somewhere else with probability (1−γ) and remains for another
period with probability γ. If an agent exits the market for exogenous reasons, she also receives a payo equalized to zero.6 Furthermore, she does not return in any subsequent
period. To keep the number of employees xed over time, (1−γ)N new agents enter the market in every period. At the end of the period, the principal can make a new oer consisting of the wage payment wt+1 to a remaining agent. If the agent accepts it, the
above procedure is repeated in the next period: the agent receiveswt+1 and chooses eort
et+1, after which the output yt+1 is realized. In each other case, i.e. if the principal does
not make an oer or the agent does not accept, both enter the matching market in the
5The case where a principal can employ more than 1 agents is considered below.
6Note that this assumption is without loss of generality even when the agent expects a positive utility
Using dP
t ∈ {0,1} to describe whether a principal is in a relationship, the payo
stream of an arbitrary principal at the beginning of a period t equals
Πt=E " ∞ X τ=t δτ−tdPτ (eτθ−wτ) #
where the expectation is over eort and wage levels, which might depend on whether the principals enters a new relationship in a period or keeps his past employee.
UsingdAt ∈ {0,1}to describe whether an agent is in a relationship, an arbitrary agent receives Ut=E " ∞ X τ=t δτ−tdAt (wt−c(et)) #
Here, expectation is overdAt an agent can leave the market for exogenuous reasons, while if she remains, she might or might not receive an oer from a principal.
We assume the following information structure. Within a match, there is no asym- metric information. This implies that eort and output can be observed by both players. Still, neither eort nor output are veriable, i.e., no explicit contract using them can be written.
All players outside a match (the market) can not observe anything that happens within.7 However, the market can see whether an agent leaves a relationship. It cannot
detect the reason, i.e., whether she left for exogenous reasons, did not receive a new oer from her previous employer, or decided to not accept an oer herself. Finally, rms can not distinguish new agents from those that already have been on the market for a longer time.
Although information within a match is perfectly symmetric, we thus have an in- nitely repeated game of imperfect public monitoring. Any employee has no information concerning the oers a principal made in the past but can use each rm's turnover history as an imperfect signal. Therefore, we follow the literature and use the solution concept of a public perfect equilibrium (PPE) in a sense that each player's actions only depend
7Whether or not the only veriable component - the wage payment - can be observed by the market
on the public history they share with the respective partner. Put dierently, the oer a principal makes to a new agent only depends on his turnover history; the same is true for the agent's acceptance- and rst-period eort-decision. When a match has been active for a while, players' actions in addition depend on the events that were observed in the relationship.8 We will later impose some additional restrictions on strategies used by
players. These restrictions simplify the analysis with having a substantial impact on the results.
Note that no additional bonus is used, and payment contingent of the agent's eort level only occurs wia the wage wt. However, this assumption is without loss of gener-
ality given our model assumptions (for an equivalent argument see Fong and Li, 2010), since bonus payments could be postponed until the next period (taking the possibility of exogenous termination into account), then becoming part of the xed wage.