The Welfare Cost of Signaling

Abstract

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. 1 A 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. 2

Two 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. 4

By 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. 5

In 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.

1. 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 $p\in (0,1)$ of being 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 $r\in (1,2)$ at home if not hired. 6 At cost $c\ge 0$, the firm can conduct a small test, with probability $q\in (0.5,1)$ 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 ${\mathbb{R}}_{+}^3$. 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:

$$c<2-r.$$

(1)

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:

$$w_H>w_L\ge r,$$

(2)

(ii) the firm maximizes profit:

$$q{w}_H+(1-q)w_L-r=f.$$

(3)

In Step 2, a type t (t= 1 or 2) potential applicant applies if and only if (iii) the wage structure is incentive compatible:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill q*max\{w_L,r\}+(1-q)*max\{w_H,r\}-r& \ge f,(\mathit{if}t=1),\hfill \\ \hfill (1-q)*max\{w_L,r\}+q*max\{w_H,r\}-r& \ge f,(\mathit{if}t=2).\hfill \end{array}\end{array}$$

(4)

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 $w_x\ge r$.

Equation (4) has the potential applicant apply if the expected value added by applying is no less than the application fee f.

From (3), $w_L\ge r$ and $w_H\ge r$ 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), $f^{{}^{\prime }}=(1-q)w_H+q{w}_L-r$ is the highest fee that induces the low type to apply; fees in the interval $(f^{{}^{\prime }},f)$ 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. 7

Note that our base model is a screening game rather than a signaling game. 8 That 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.

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).

2. 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. 9

The hiring game is played in sequence as follows:

• The headhunter sets a fee f.
• The applicant decides whether to pay the fee to enter the market.
• The firms quote wages.
• If she/he has entered the market, the applicant chooses a firm and applies.
• 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. 10
• 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.

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.

3. 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 assuming $w\ge f\Rightarrow \frac{\partial U}{\partial (w-f)}(w-f,f)$ is decreasing in $w-f$ for any f.

Let $w>r$; there exists an $ϵ>0$ small enough such that $w-ϵ\ge r$. Then, the wage/fee schedule $w-ϵ$, $w+\frac{1-q}{q}ϵ$ 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. 11

Naturally, assume the risk premium increases with ϵ. 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 ϵ into the amount of risk premium that makes the high type indifferent. Then, we have the following corollary:

Corollary 2 (Third-party corollary).

$s^{*}(ϵ)=[w-ϵ,w+\frac{1-q}{q}ϵ,f-RP(f,ϵ)]$ 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 ϵ. 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 $f-RP(f,ϵ)$, 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.

4. 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:

$$w_{H}q+w_L(1-q)-f=2-c,$$

(5)

$$w_H(1-q)+w_{L}q-f<r.$$

(6)

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:

$$(2q-1)(w_H-w_L)>2-c-r.$$

(7)

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.

5. 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. 12

Seeking 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. 13 The firm maximizes the payoff:

$$E[2-c+m_{n}f-q{w}_u-(1-q)w_d\mid n,m\ge 1].$$

(8)

This can be simplified to (ignoring the constant $2-c$):

$$E[m_n\mid n,m\ge 1]f-q{w}_u-(1-q)w_d.$$

(9)

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:

$$E[\frac{w_d}{m_{n-1}+1}+\frac{q(w_u-w_d)}{m_{n-1}+1}-f\mid n]=r,$$

(10)

or:

$$f=E[\frac{1}{m_{n-1}+1}\mid n]*(q{w}_u+(1-q)w_d)-r.$$

(11)

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.

6. 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. 15 A 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.