The Welfare Cost of Signaling

Might the resource costliness of making signals credible be low or negligible? Using a job market as an example, we build a signaling model to determine the extent to which a transfer from an applicant might replace a resource cost as an equilibrium method of achieving signal credibility. Should a firm’s announcement of hiring for an open position be believed, the firm has an incentive to use a properly-calibrated fee to implement a separating equilibrium. The result is robust to institutional changes, outside options, many firms or many applicants and applicant risk aversion, though a sufficiently risk-averse applicant who is sufficiently likely to be a high type may lead to a preference for a pooling equilibrium.

Adverse selection becomes a concern when a party A faces a decision based on information possessed by a party B, whose utility is also affected by A's decision.That is, under what circumstances can party A rely on information communicated by party B?
Spence's 1973 paper [1] introduced a model in which employers may use education as a screening device.In a related context, Akerlof (1970)  [2] provided perhaps the most widely-taught adverse-selection example.Stiglitz (1975)  [3] discussed the concept of screening in the context of employment and education.All of the above mechanisms are costly ways of solving an adverse-selection problem by creating an incentive to self-select.By contrast, in cheap-talk games (Crawford and Sobel, 1982 [4], Chakraborty and Harbaugh, 2010 [5] and the references in the latter), where communication is privately and (probably) socially costless, information that can credibly be transmitted is limited, usually severely.This paper asks: Since a sender must incur a cost of transmitting if the message is to be credible (for present purposes, the cost of obtaining, say, an MBA degree, is here labeled a cost of transmitting), to what extent can the cost be reduced for society by using a transfer instead of a pure resource cost?
We address this question not to explain common occurrences in markets, but to better understand the foundations of the economics of transacting under asymmetric information.To explore these foundations, suppose that a firm can credibly commit to considering only those applicants who pay an application fee that might be substantial. 1A test that might distinguish between applicants in some aspect of their suitability could still be conducted, but only if the firm's resource costs of administering the test and evaluating the effectiveness shown are quite small and an applicant's resource costs of preparing for and taking the test are negligible compared both to the resource costs of a usual signal and to the size of the application fee (the firm comparing a privately known threshold to the applicant's credit score, or her/his driving record, or her/his number of semesters on the Dean's list or quickly searching Facebook or Instagram or YouTube for incidents).With this setup, the question becomes whether a suitably calibrated application fee can achieve the same types of signaling equilibria that are accomplished by calibrating the resource cost of the usual sort of signal, such as obtaining a particular level of education.That substantial application fees may not be common in labor markets of acquaintance does not bear on the relevance of this research. 2wo papers touch on this question, both less directly and subject to objections.Wang (1997) [6] introduces an employment model in which only if the firm commits to a wage schedule before the applicants pay the fee might an application-fee equilibrium be possible.Though set-of-wages, positive-application-fee equilibria may be possible below, this possibility need not require the firm to commit to a wage schedule before an applicant decides whether to pay the fee (cf.Section 2).As to using the necessity of commitment to explain why no application fee is observed in reality as Wang (1997)  [6] does, no practical reason can be given why a firm could not commit to a schedule of multiple wages corresponding to multiple estimated productivities (indeed, this is a feature of nearly every job posting seen in the market for Ph.D. economists).Furthermore, the pre-commitment argument is based on the assumption that firms have full control over wages.If the wage is instead determined through, say, alternating-offer bargaining, 3 it is obvious that applicants can still expect to attain some surplus, making a positive application-fee possible.
Guasch and Weiss (1981) [7] suggest that applicants' risk aversion and an assumption that applicants do not have perfect information about themselves may prevent a positive-application-fee equilibrium.As shown below, risk aversion alone is insufficient to prevent a positive-application-fee equilibrium.The Guasch/Weiss model [7] requires the assumption that the labor-supply constraint is not binding, which is problematic: If there are more than enough high types applying, why test all of them?Where do the "extra" high-type applicants go?Firm profit maximization implies that they are not hired, while applicants' expected returns show that they get paid and hired. 4y assumption, the firm genuinely wishes to hire someone, and this is believed by the applicant (the credibility of many firms, especially prestigious ones, is in fact too valuable to risk fraudulently collecting application fees for nonexistent jobs).
Although the models use job-application settings, to varying degrees, they can apply to other contexts, as well.For example, the job vacancy can easily be interpreted as a promotional opportunity within a firm.The model may extend to payment below productivity during a required internship period.Another possibility is that a firm may attempt to signal credibly the quality of a product or product line or a service by a donation to a charity that it knows will be publicized at no cost to the firm. 5n the following model and variations, a separating equilibrium always exhibits a positive application fee.Whether there is a separating equilibrium depends entirely on the firm's incentives (or the firm's and headhunter's incentives in Section 2).Should no separating equilibrium exist, the firm's only options are not to hire or to charge no application fee and hire any applicant without testing.With testing cost sufficiently low, separating equilibrium almost always exists.

The Base Model
Consider a game between a profit-seeking monopsony employer and a potential applicant, both risk neutral.The applicant is either Type 1 or Type 2, and knows her/his own type.The firm does not know the type, but correctly knows the distribution of types (probability p ∈ (0, 1) of being 2 Among many markets with adverse-selection issues exhibiting application fees are college admissions, scholarly journals and nightclubs with live music.

3
Cf. van Damme, Selten and Winter (1990)  [8] and references there.4   The models used in this paper are similar to those of Guasch and Weiss (1981)  [7].In fact, the one-firm, multiple-applicant case (Section 5) can be regarded as a simpler version of their model, while avoiding the "labor-supply constraint not binding" problem.

5
For a context yielding several more examples, see Spence (2002)  [9].Type 1; 1 − p of being Type 2).Type t worker has productivity t if working for the firm (t = 1, 2).Both types can produce r ∈ (1, 2) at home if not hired. 6At cost c ≥ 0, the firm can conduct a small test, with probability q ∈ (0.5, 1) of correctly revealing the applicant's type and probability 1 − q of being misleading, thus possibly of very little reliability.The hiring game is played as follows: Step 1 The firm chooses its strategy s = (w L , w H , f ), where w L is the wage offered to an applicant with Test Result 1, w H is the wage offered to an applicant with Test Result 2 and f is the application fee.
Step 2 The potential applicant sees the wage/fee schedule s and decides whether or not to apply for the position.If she/he applies, she/he must pay the application fee.
Step 3 If the applicant has applied in Step 2, she/he takes the test and the result is revealed for both the firm and the applicant.The applicant then decides whether to accept the wage offer fitting the test result.
For the above defined game, the firm's strategy space is R 3 + .The applicant chooses (App( f , w L , w H , t), Acc( f , w L , w H , t, x)), in which t is her/his type and x is the realized test result."App" can be either "apply" or "not"; "Acc" can be either "accept" or "not".
To avoid trivialities, assume: That is, the test cost cannot be so large that the firm would not make an offer to a known high type.For simplicity, also assume that the applicant accepts the offer if she/he is indifferent in Step 3 and that she/he applies if she/he is indifferent in Step 2. Assume, of course, that both the firm and the applicant play to maximize their expected payoff.The above specifications yield the following theorem, proven in Appendix A.

Theorem 1 (Main theorem).
A strategy profile is a subgame-perfect equilibrium of the above-defined game if and only if: In Step 1, the firm implements a separating equilibrium in which the potential applicant applies if and only if she/he is a high type and hires anyone that applies while setting w L , w H and f , such that (i) the firm makes an acceptable offer: (ii) the firm maximizes profit: In Step 2, a type t (t= 1 or 2) potential applicant applies if and only if (iii) the wage structure is incentive compatible: In Step 3, the applicant accepts the offer if and only if the wage is no less than r.That is, for an applicant with test result x, accept if and only if w x ≥ r.Equation (4) has the potential applicant apply if the expected value added by applying is no less than the application fee f .From (3), w L ≥ r and w H ≥ r ensure that if the applicant applies, she/he is hired, at a test-dependent wage level.w H > w L separates the value of applying for different types, in favor of the high type, given q > 0.5.
The fee determined by Equation (3) leads the high type to apply, though indifferent.Any higher fee prevents the high type from applying.The low type does not apply because, compared to the high type, she/he has a lower chance of receiving the high wage, but faces the same application fee.For given w L , w H satisfying (2), f = (1 − q)w H + qw L − r is the highest fee that induces the low type to apply; fees in the interval ( f , f ) reduce fee income and lead the high type to strictly prefer applying, without otherwise affecting the outcome.
For example, setting w L = r, w H > r and letting f be determined by Equation ( 3) yield a subgame-perfect equilibrium.In such an equilibrium, both types are in their most productive positions (firm for high types and home for low types), and perfect separation is achieved without testing the low type, thus saving on testing cost.The application fee serves to make an imperfect sorting device (the test) perfect, even though the fee is purely a private cost rather than a social cost.
This welfare reassessment, that separating equilibria can be achieved by having the signal's cost be a transfer, rather than a resource cost, is robust to institutional changes.The online supplement extends the analysis to institutions in which: (i) the applicant is freely considered, but only if she/he has paid a fee will the firm observe the test result; (ii) the fee must be paid for a positive (but possibly quite low) probability that the firm will observe the test results (and otherwise, the firm does not incur a testing fee); and (iii) the fee must be paid for a positive probability that the firm will observe the test results, but with the fee refunded in the random event that the firm does not observe the results. 7ote that our base model is a screening game rather than a signaling game. 8That is, the firm makes all of the decisions first and then lets nature and the applicant do all of the separating, rather than observing some signals sent by the applicant and then making decisions based on updated information about the applicant type.Note also that, while the fee is an effective screening device, it cannot be made into a signaling device simply by moving the wage decision to the last step.Subgame perfection would require that the firm pay no more than r in the last stage, and as a result, no applicant would pay a positive fee to apply.A variation in which the firm's wage decisions occur after a potential applicant has decided whether to pay an application fee is analyzed in the next section.
Comparison with Spence (1973) [1]: Spence's model differs from ours in two ways.His costly signal is an effort that per se serves no economic or social purpose, but is be assumed to create significantly less disutility for a high type than a low type (the same differential capabilities that make a high type a more productive employee are assumed to yield the lower disutility of the communicative effort).In this model, the signal cost is a transfer, and it would be untenable to assume that the high type had a significantly lower marginal utility of income.The monetary nature of the signal carries the social-welfare advantage that the money may be put to an equally productive purpose.It carries the disadvantage that paying the application fee generates the same disutility for the high type and the low type, so paying the application fee cannot by itself credibly signal type.Thus, we add to Spence's model an inexpensive test with an informational content that may be nearly meaningless (e.g., the fraction of the courses taken during her/his senior year that are numbered by her/his university as senior-level courses), but simply is more likely to be passed by a high type than a low type.Now, setting test-result-dependent wages so that the high type is nearly indifferent over paying the application fee separates: a lower likelihood of a passing test result leads the low type to see an insufficient expected advantage to paying the same application fee.

7
The latter two institutions require the firm in Step 1 to set two additional wage levels: that offered to an applicant who paid the fee, but whose test result was not observed, and that offered to an applicant who declined to pay the fee.
For a discussion of the difference between signaling and screening games, see Sections 13.C and 13.D in MasColell, Green and Whinston (1995) [10].
Spence's signal is often described as obtaining a university degree at a job-appropriate level (e.g., MBA), which if not justified as a human-capital investment, might be seen as very costly to a high type and, thus, to society.Spence does not himself so limit his model; the signal in the right circumstances might be exceeding a carefully-chosen threshold on a standardized test, if attaining the threshold yields sufficiently high disutility for a low type.Thus, in some circumstances, Spence's separating equilibrium might have a high type incurring such a small cost transmitting a signal (that would have been highly costly to a low type) as to approximate "first-best": very low social costliness of the signals for types that actually transmit them.In this paper, the approximate first-best simply comes from the lack of social cost of the signal no matter how high the private cost to the signaler.
In the theorem above, the possibility of approaching a low private cost can be attained similarly to Spence's model.The f in the theorem must be positive, but can be made arbitrarily small, adjusting to w L = r, lowering w H to satisfy (4).

Application Fee in a Signaling Game: Adding a Third Party
As discussed in Section 1, the base model's application fee cannot be shifted directly to a signaling device.Suppose there is perfect competition by firms hiring in this labor market, but an applicant can only be considered by a firm after she/he pays a fee to that particular firm.Once an applicant has paid the fee to a particular firm, that firm no longer faces any competition in hiring that worker and so offers at most wage r.Any positive fee is then impossible.
This issue may be resolved by involving a third party.The applicant must pay a fee to this third party to enter the market; upon entry, all firms in this market can compete for her/his employment.
Consider a job market with multiple firms competing with each other, while still only having one applicant, with type assumptions as before.Now, assume there is a headhunter, who holds some monopoly power in the market: firms can only hire a job applicant through the headhunter, who may demand a fee for the applicant to be available for hire. 9he hiring game is played in sequence as follows: 1.The headhunter sets a fee f .2. The applicant decides whether to pay the fee to enter the market.
3. The firms quote wages.4. If she/he has entered the market, the applicant chooses a firm and applies. 5.The applicant is tested, costing the firm c; she/he signs a waiver ceding the right to apply to or negotiate with any other firm. 106.The applicant and firm learn the test results; previously-set wages are offered to the applicant; she/he decides whether to accept or not; if she/he accepts, she/he is hired.If not, she/he returns home and produces r.
The waiver is a convenient way to: (i) keep both w L and w H wage quotations relevant to applicant decisions; and (ii) prevent the applicant from applying to another firm if she/he tests low at the current firm.It yields the most straightforward comparison to the base model.
For a natural choice of tiebreakers, the main result extends sensibly: Corollary 1 (Third-party corollary).For the headhunter to set f = 2 − r − c, firms to set (w L , w H ) so that w H > w L and (3) and (4) are satisfied, high types to apply and the low types not to apply constitute a separating equilibrium.

9
A frequent example is a government agency that has to specify that an applicant meets certain criteria before she/he can be hired into a particular field or for a particular job.This model considers the agency setting the fee in excess of its cost of certification. 10 Having the headhunter cover the testing cost out of fee revenue only yields obvious adjustments to the equilibria.
The setup and proof are in Appendix B. Note that this result does not require a competitive industry seeking to hire a high type: suppose there is but a single firm who nonetheless can only hire an applicant who paid the fee to a headhunter; if the headhunter sets a fee as above, then the firm optimally sets wages (w L , w H ) satisfying ( 2), ( 3) and ( 4), generating a separating equilibrium.

Risk-Averse Applicants
This section returns to a single (risk-neutral) firm, to consider applicant risk aversion.Guasch and  Weiss (1981)  [7] noted the obvious: a risk-averse applicant requires an expected return greater than r to accept the risk of an uncertain test result implying an uncertain wage.Intuitively, if the risk premium required is high enough and the cost of hiring a low type is low enough, the firm may be unwilling to pay the risk premium as the cost of separating equilibrium and hire everyone without testing instead.Assume both types of applicant have the same pattern of risk tolerance.
Under what conditions can a separating equilibrium be preserved?Instead of positing a particular risk-averse utility function, consider a wage/fee schedule and ask how high a risk premium is needed for a high type to accept.Specifically, in the base model, the firm may set the two wages arbitrarily close; the focal issue is the risk premium required if w L is close to w H .
Above, as is usual, the applicant is indifferent between a wage of 12 and fee of two and a wage of 13 and fee of three.However, this section's analysis of risk aversion is clarified by generalizing to a utility function U(w − f , f ), for either type of applicant, with the usual concavity maintained via ) is decreasing in w − f for any f .Let w > r; there exists an > 0 small enough such that w − ≥ r.Then, the wage/fee schedule w − , w + 1−q q and f = w − r is a viable schedule to implement separating equilibrium in the risk-neutral case.As above, q is the probability the test correctly identifies the applicant's type.Since this involves risk, a risk-averse high type would demand a risk premium to accept such an offer; for clarity, treat the risk premium as being subtracted from f . 11aturally, assume the risk premium increases with .Therefore, the firm would prefer to offer wages as close to each other as possible.
A schedule s that makes the high type indifferent over accepting would not be accepted by the low type, who would end up receiving the low wage with a greater probability than the high type.Therefore, the firm only has to make sure that high types are indifferent in order to implement a separating equilibrium.
Let RP be a function that maps f and into the amount of risk premium that makes the high type indifferent.Then, we have the following corollary: ] is a potential schedule in a separating equilibrium.The applicant types separate under this wage/fee schedule, provided the firm is willing.
Note that all possible sets of wages satisfying Equation ( 2) can be represented by the above wage schedule, via changing .Since any separating equilibrium must satisfy Equation ( 2), the wage schedule can be represented as above.Given the above wage schedule, the fee must be f − RP( f , ), as the high type would not accept any higher fee, and the firm's profit is suboptimal for any lower fee.Therefore: Proposition 1.Any separating equilibrium takes the form described in Corollary 2.
Appendix B delves into risk aversion in more detail, finding: a minimal assumption for separation to occur, separation impossible with truly catatonic risk aversion and how unusually risk aversion must be modeled for pooling possibly to be preferred.

Multiple Firms Competing for the Applicant
This section examines whether multiple firms competing for one risk-neutral hire can affect the realization of separating equilibrium.If separating equilibrium is still achievable, it must be allowing the high type to get all of the surplus, since otherwise, another firm would offer a higher wage and attract the high-type worker away.On the other hand, the low type must be expected to get less than r if she/he were to apply.This immediately means that any wage/fee schedule that implements separating equilibrium with multiple firms needs to separate the two types sufficiently far.Opportunities to separate with w L closely below w H are more restricted, perhaps preventing separating equilibrium were this section blending firm competition and applicant risk aversion.
For separation, the wage/fee schedule must satisfy: Equation ( 5) ensures the high type's expected wage minus fee equals social surplus; Equation (6) discourages the low type from applying.Subtracting ( 6) from (5) yields: Equation ( 2) is still needed to ensure hiring all that applied.Corollary 3. Any set of w H , w L and f that satisfies Equations (2), ( 5) and (7) yields a wage/fee schedule for a separating equilibrium.
Details and a proof are in Appendix B.

One Firm, Finitely Many Potential Applicants
Instead of one potential applicant, suppose there are n; each is independently a low type with probability p; n and p are assumed common knowledge.The firm can only use one worker productively. 12eeking a separating equilibrium, the firm has neither the desire, nor the need to test all applicants.Let it adopt the strategy of testing one randomly-selected applicant, hiring her/him if her/his test result is high, and otherwise hiring a second randomly-selected applicant (possibly the same applicant as the first) without conducting even a second test.
This testing strategy can support a separating equilibrium.If there is no high type in the pool, no one applies, and the firm receives no profit.As long as there is at least one high type, there are fee-paying applicants; the firm tests and hires someone.Therefore, the firm maximizes expected payoff conditioning on at least one high type in the pool.
Let m n be the realization of the number of high types in a pool of n, w u be the wage offered to the applicant getting high test results and w d be the wage offered to the applicant selected through the second random draw. 13The firm maximizes the payoff: This can be simplified to (ignoring the constant 2 − c): A high type, competing with n − 1 rival potential applicants who are each a low type with probability p, though indifferent, will apply if facing a wage/fee schedule satisfying: or: Facing the same wage/fee schedule, a low type has the same expected payoff if randomly selected second, but a lower payoff if randomly selected first, as the probability of testing high is less.Therefore: Corollary 4. The firm can implement a separating equilibrium with any wage/fee schedule that satisfies (2) and (11).
Appendix B details how the base model is a special case of this model.Adding an option to hire without testing, in a pooling equilibrium, the firm would simply hire the first of the applicants at wage r without testing.Unlike the introduction of more firms, introducing more applicants does not seem to qualitatively change the feasibility of using f and a test to create separating equilibria.

Discussion
The models presented yield the following conclusions:

•
It is possible to use a transfer to implement a separating equilibrium; in that sense, the private cost of signaling need not be a social cost.

•
Commitment to a wage by the firm is not necessary to use a transfer as a signaling device.14• Applicant risk aversion alone is normally insufficient to prevent the existence of a separating equilibrium.Considerable differences in home productivity across types may increase the likelihood that equilibrium requires pooling.
An assumption of the base model is that the test cost c is nonnegative. 15A negative c may make it optimal for the firm to test everyone (the last inequality in Case 3 of the proof may not hold if c < 0).For some situations, applying this model would naturally suggest a negative c.If the test represents some form of internship or other productive activity and the fee as the reduced pay in this activity, there is a legitimate reason to claim that c can be negative, meaning the interns are producing more than the funds it took the firm to set up such a program.
Similar to the discussion about the internship, if an employee's type may be (imperfectly) revealed only after some periods of employment, the employment period before such a revelation can be considered a test to determine the wage afterwards.Aside from the possibility that c is negative, there are two differences to the base model: the "test" cost c now is less for a high than for a low type, and the "test" is possibly perfect.There will still be a set of separating equilibria if c is still positive.
Are results affected if the firm has to spend money advertising jobs in order to attract applicants?Add to the base model an assumption that the firm needs to incur a fixed cost in order to let the applicant be aware of the opportunity, i.e., to apply for the job.However, once paid, it becomes a sunk cost, so it should not affect the firm's choice of wage and fee.It can affect the firm's choice of whether to enter the market.
Appendix C shows that if instead of having two potential applicant types, types are continuously distributed along an interval on the real line, the separating equilibria are not affected if the reservation wage remains constant across types.
Two main differences exist between Guasch and Weiss (1981) [7] and the models in this paper so far: the models so far allow hiring of an applicant with a low test score and a universal reservation wage across types.What will be the impact of relaxing the latter restriction by introducing a variable reservation-wage based on type to these models?In Appendix D, variable reservation wages are introduced to the base model.Appendix F provides proofs and extends to multiple firms or multiple applicants.It turns out that while variable reservation wages may yield a smaller set of wage/fee schedules implementing separating equilibrium (by requiring a minimum difference between w H and w L ), firms still maintain the option to separate.For models in which a pooling equilibrium is possibly optimal, the firm makes the same choice between separating and pooling as if the reservation wage is a constant.

Case 1: Only High Types Apply, Only the High-Result Applicant Is Hired
All such possibilities can be dealt with as if w H ≥ r > w L .Then, adjusting (3), the highest fee acceptable for a high type to apply becomes: The equality yields high-types applying though indifferent, the inequality low types not applying.Compared to s * , the firm's profit has fallen by (1 − p)(1 − q)(2 − r), as high types who tested low were profitably hired in s * .Reducing the fee to f < f * * at best allows increasing w H by ( f * * − f ) q , which cannot yield an increase in expected profit, so offering no advantage.Same can be argued for increasing the fee to f > f * * .
Case 2: All Types Apply, All Are Hired An applicant is always tested (as required in the base model).Initially assuming w H ≥ w L ≥ r, sets the highest acceptable fee to f * * = (1 − q)(w H − r) + q(w L − r).With both types hired, expected productivity is 2 − p, so expected profit is at most 2 Again yielding lower expected profit than s * .

Case 3: All Types Apply, Only a High-Result Applicant Is Hired
Hiring only an applicant who tests high, as in case 1, it suffices to consider w = w H ≥ r > w L .However, to get the low types to apply, the highest fee becomes f * * = (1 − q)(w H − r), which has the low type apply though indifferent (and the high type strictly prefer applying).As no type offered wage w L accepts, expected profit at f * * is where the term in [] is productivity less wage for a high type who tests high, that in {} is the same difference for a low type who tests high, the first equality substitutes for f * * , the second collects terms in w H .As q > 1 2 , the coefficient of w H is negative, so expected profit is maximized at w H = r, which sets f * * = 0. Substituting these values of w H and f * * into the left-hand side of (A3) yields , where dropping the nonpositive term in provides the weak inequality, and substituting the larger 1 for q and the smaller (1 − p) for 1 provides the strict inequality, again yielding lower expected profit than s * .
Of course, it does not matter to the applicant to whom she/he pays the fee.Therefore, if Equation (4) from section I holds, the high-type applicant continues to apply though indifferent, and the low type continues to strictly prefer not applying.
In a separating equilibrium, each firm for whom the probability of receiving an application is positive must set (w L , w H ) so that w H > w L and (3) is satisfied.Despite introducing the headhunter, (3) still makes the high type indifferent over entering the market and applying, and yields a strict preference for the low type to stay home.
Suppose the headhunter sets f ≤ 2 − r − c, the high type applies, and the low type may or may not apply.Were a given firm to face a pattern of wage offers in which every other firm offered the high type an expected wage below 2 − c, its best response would be to offer slightly higher wages.
Thus, for the headhunter to set f = 2 − r − c, firms to set (w L , w H ) so that w H > w L and ( 3) and ( 4) are satisfied, high type to apply, and the low type not to apply, constitutes a separating equilibrium.In such an equilibrium, the headhunter expropriates all the social surplus, and the high-type expected wage is 2 − c, leaving firms with zero profit in this labor market.The high type applies to the firm whose wage offer has the largest difference w H − w L , but the expected wage being driven up to productivity removes any incentive for other firms to deviate to attract the high type.
Are these the only equilibria?The headhunter can be shown, without doing algebra, to set too high a fee to allow a pooling equilibrium in which both types enter.In separating equilibrium, the headhunter extracts all surplus of an efficient labor market.The low type only enters if the chance of being hired justifies paying the fee, and hiring the low type reduces surplus.Wages yielding a high enough expected wage to yield low-type entry must pay some surplus to the high type, who has a greater probability of being offered w H . Therefore, the headhunter would receive only a portion of the smaller surplus.Details are provided in Appendix G.
If application-fee revenue is used by the headhunter for some social purpose with social marginal valuation approximately dollar-for-dollar (or better), then the applicant's private cost of signaling is nearly a transfer, at most a negligible social cost. 16

Appendix B.2. Risk-Averse Applicant
First consider, for later comparison, catatonic risk-aversion: the applicant would always value a lottery at the lowest possible payoff.For this case, RP( f , ) = r − (w − ) + f = ,17 the upper bound of the RP function.RP = guarantees a payoff of at least r, even if the test result suggests a low type.Therefore, this case yields the low and high types evaluating the wage/fee schedule identically, preventing a separating equilibrium.
In less extreme cases, the firm can choose f to minimize RP.Suppose: this is equivalent to a continuous utility function at r.A separating equilibrium can be implemented by choosing the right f and setting as close to zero as possible to pay almost no risk premium.This is different from Guasch and Weiss (1981) [7], because in their model, an applicant with low test result is not considered for hire (equivalently, w L is forced to be less than the reservation wage r), thus selecting w L and w H arbitrarily close is not an option.In addition, their model also assumes that the reservation wage varies with type, which prevents w L and w H from being arbitrarily close even if the restriction on w L was relaxed. 18ow must one model risk aversion so that the firm might prefer a pooling equilibrium?Assume a strictly positive risk premium for any distance between w H and w L (less extreme than catatonic risk aversion).Let: i.e., for any f , applicant utility function is discontinuous at net income r. 19Instead of production of 1 and 2, consider production level l for low type and h for high type.Compare firm's surplus in the separating equilibrium and in the pooling equilibrium in which the firm hires both types without testing.
The firm's surplus in separating equilibrium is: and in pooling equilibrium is Subtracting (B4) from (B3): The only seeks a separating equilibrium if expression (B6) is positive.The base model assumes 1 < r to give a welfare motivation to not hire the low types.With a similar assumption that l < r, separating equilibrium becomes more likely as p goes up (the low type becomes more likely, so hiring without testing becomes more harmful), as c or z goes down (cost of separating equilibrium becomes lower), as l goes down (the cost of hiring the low type becomes more harmful), as r goes up (hiring becomes more costly).Note that h is not in expression (B5), since in both separating and pooling equilibria, high types are hired.
This analysis applies anytime a strictly positive risk premium is required.For example, if Equation (B1) holds, but for any reason cannot approach zero, that is, w L and w H cannot be arbitrarily close, a risk premium bounded above zero may be needed.A possible reason for not to approach zero is at-home productivity r differing with type.See Appendix D.

Appendix B.3. Multiple Firms
A separating equilibrium can be achieved unless there is an arrangement that can provide an expected wage minus fee for high types higher than 2 − c, while keeping the firm's return non-negative.Since the firms and the low type are already getting their reservation level, allowing the high type to get even more requires higher social surplus than separating equilibrium can attain.Since the only deviation from the full-information optimum in the separating equilibrium comes from testing the high types, if randomizing tests are disallowed, the only possible way to achieve higher surplus is by hiring everyone without testing, which can be checked by: though indifferent.Equation (C4) (which implies (C2)) minimizes the surplus paid to types above r + c, allowing the firm the maximum attainable surplus.Note that the distribution of types does not enter the equations characterizing separating equilibrium.
To consider pooling equilibrium, enable the firm to hire an applicant not taking the test.A pooling equilibrium is implemented if the firm hires everyone without testing at wage r.Equilibrium profit depends on the distribution of types.
A direct comparison of a separating equilibrium (where the hiring of an applicant not taking the test is disallowed) and a pooling equilibrium (where such hiring is allowed) would be unconvincing.Therefore, consider a separating equilibrium assuming any type applicant can decline to pay the fee in order to take the test, and may still be hired.Interestingly, an equilibrium schedule s leads to an interval of types applying and choosing to take the test, and the firm does not make an offer to non-test-taking types.To see this, suppose the firm hires the applicant even if test-taking is declined.In equilibrium, there must exist a type t * < 2 such that: (i) types t ≥ t * apply, take the test, and are hired at wage w H if testing high, w L if testing low; and (ii) types t < t * apply, decline to be tested, and are hired at wage r.If t * ≤ r, the firm is better off not hiring non-test-takers, so an equilibrium requires t * > r.However, in this case, all types are hired, all are paid at least r, and the firm incurs test-administration cost (2 − t * )c > 0, so it is strictly better off pooling than separating and hiring non-test-takers.
Therefore, the separating equilibrium specified by Equations ( 2), (C3) and (C4) still stands.The firm compares the loss of hiring types below r + c at wage r (could be negative depending on distribution of types) with the cost of testing types above r + c, if the former is greater the firm implements a separating equilibrium, otherwise it implements the pooling equilibrium. 21ow discard the assumption about the smallest monetary unit δ, and the possibility of hiring a non-test-taking applicant.Suppose types are uniformly distributed between one and two, and let r(t) be a continuous increasing function distributed on [1, 2], with r(1) > 1 and r(2) < 2, so that the firm still might profitably hire type 2, but not type 1.Furthermore, assume the firm is again not allowed to hire an applicant not taking the test.
In Figures C1-C3 below, the thin line, y(t) = t, represents the gross productivity for each type.The dashed line shows net productivity, productivity reduced by test-administration cost c, a downward shift (0.5 in the graph).The thick line represents r(t).By changing s = (w L , w H , f ), the firm can generate as W(t | s), net expected wage, any line with nonnegative slope, obtaining the employ of any types for which W exceeds r, profiting by the height (net productivity -W).
These figures offer some interesting cases.Figure C1 illustrates the action the firm takes if r(t) is a linear function of t, with r(1) > 1 − c and r(2) < 2 − c.In this case, matching W(t | s) = r(t) is the equilibrium, with the firm obtaining all the surplus (the shaded area).Figure C2 illustrates how a possible optimum of the firm might not involve hiring the highest types.This time r(t) is flat until t = 1.8, and then steep.A possible W22 is the dotted line in the figure, which yields the shaded area as surplus for the firm, while the triangle below the shaded region is applicant surplus.An interval of intermediate types is hired, while lower types are insufficiently productive, higher types overly expensive.Figure C3 shows a case in which c is so high that testing guarantees a loss, thus neither testing nor hiring occurs.Were hiring without testing possible, the firm can separate potential applicants without using the test or wage differentials.A flat wage can separate due to differences in reservation wage across types.In the case shown however, the firm does not wish to do so and prefers not to hire at all instead.( (2,0) type productivity-c r probability m.Now, however, suppose the application fee is refunded unless the test is actually administered.The firm can again approach the full-information optimum, as in (ii) by setting w H > w L ≥ r > w N , w M ≥ r and so that high type pays the fee though indifferent, and let m approach 0.24 Appendix E.1.Details of Separating Equilibria Here, w M is used to denote the wage for someone who signs up for the test but not receiving a test, w N is for someone not signing up for the test.
For (ii), to achieve separation, the firm makes the high type indifferent over paying the application fee: The inequality ensures that the low type does not pay the fee, and is achieved given (2).Conditioning on separation, the firm wants to hire the fee-payer for sure, so w L , w H and w M are all no less than r.Even if the firm is allowed to hire a non-fee payer, it does not wish to do so, which yields w N < r.The fee achieves separation.Then firm's profit is: with equality due to the expected wage being r + f [from (E1)].As m goes to zero this approaches the full-information optimum, so a pooling equilibrium can never be more profitable (even allowing w N as in (i)).For (iii), the separating condition becomes: and the firm's profit becomes: This, again, approaches the full-information optimum.

Appendix F. Details for Variable Reservation Wage
Base model: A firm can always find such a wage/fee schedule by first choosing any pair of w H and w L satisfying Equation ( 2) and with a wide enough difference to satisfy equation (D3).Then calculate the fee using Equation (3).Therefore, a separating equilibrium is achievable by the firm.
In order to check the optimality of such a separating equilibrium for the firm, compare the variable reservation wage model with the one if the reservation wage for both types is r H .The separating equilibria in both cases yield the same return for the firm: the potential applicant is only tested and hired if she/he is a high type, and the firm claims all the surplus.Appendix A shows that in the later case, a separating equilibrium is optimal for the firm.Therefore a separating equilibrium is also optimal in the former case if there is no way for the firm to take advantage of a reduced cost to hire a low type.Indeed, in a separating equilibrium, since the firm hires only if the potential applicant is a high type and receives all the surplus, the only possibility to improve comes from hiring the potential applicant if she/he is a low type as well and receive surplus from (a) the production of the low type or (b) savings on the cost of conducting the test.Since the low type reservation wage is still greater than her/his productivity, the surplus from (a) is negative.As for (b), the assumptions in the base model require any hired worker to take the test, preventing the firm from saving on test cost.Therefore, for this model, it is optimal for the firm to implement a separating equilibrium.
(i): This adds the possibility of a pooling equilibrium in which the firm hires regardless of the type without testing.However, compared to the case of a single reservation wage r H , the firm does not change its behavior under variable reservation wage.Earlier in this section, separating equilibria yield the firm the same payoff in both cases.In order to implement a pooling equilibrium, the firm needs to pay a wage of r H regardless of the type, same in both cases.Therefore the variable types model has the same solution (either separating equilibria or pooling) as the one in which both types having a reservation wage of r H .
Multiple firms: If the variable reservation wage assumption discussed earlier is added to the model with multiple firms, Equation (6) needs to be adjusted accordingly: Subtracting this from Equation (5) yields: This is comparable to Equation (7).The rest of the analysis remains the same as in Section 4. Therefore, the solution to the multiple firms competing for an applicant model with variable reservation wage is the same as if the reservation wage for both types is r L .
Multiple applicants: Apply the same variable reservation wage assumption to Section 5. Equation (10) changes to Additionally, Equation (11) changes to: | n] * (qw u + (1 − q)w d ) − r H . (F4) In addition, to discourage low types from applying, the following must be satisfied: Subtracting this from Equation (F4) yields: which can be rearranged into: Equation (F7) specifies how much of a wage difference is required in order to implement a separating equilibrium, similar to Equation (D3) in the base model.The firm implements a separating equilibrium by choosing a wage/fee schedule satisfying w u > w d ≥ r H , (F4) and (F7).
In the pooling equilibrium, the first of the applicants is hired at wage r H .The firm chooses a separating equilibrium or the pooling equilibrium based on its expected payoff.

Figure C1 .
Figure C1.Under linear r(t), the firm maximizes profit by matching W(t | s) = r(t).

Figure C2 .Figure C3 .
Figure C2.If types are continuous and the reservation wage is variable, the firm may only hire intermediate types.