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Strategic dropouts
Games and Economic Behavior 50 (2005) 79–88 www.elsevier.com/locate/geb
Strategic dropouts Ram Orzach a,b,∗ , Yair Tauman c,d a Wayne State University, USA b Tulane University, USA c Tel Aviv University, Israel d SUNY, Stony Brook, USA
Received 10 September 2003 Available online 8 December 2004
Abstract Following Spence, this note provides an education signaling model to explain the phenomenon of gifted entrepreneurs who acquire less education than ordinary individuals. Two types of individuals, ordinary and gifted, are considered. Each one of them can either convince an investor to fund his enterprise or approach a competitive job market. The probability that an ordinary individual succeeds to establish a successful enterprise is smaller than that of a gifted individual irrespective of his education level. The probability of an ordinary individual succeeding increases with the level of education. In a separating equilibrium gifted individuals curtail their eduction to a level below that of the ordinary ones. This happens if the value of a successful enterprise per dollar investment is sufficiently large, on the one hand, but not too large to guarantee that the expected value of an enterprise run by an educated ordinary entrepreneur falls below the investment cost, on the other. 2004 Elsevier Inc. All rights reserved. JEL classification: C72; D82
1. Introduction Spence’s (1974) signaling education model focuses on the phenomenon of a high-ability individual investing significantly in education to a level just sufficient to deter a less able * Corresponding author.
E-mail addresses: [email protected] (R. Orzach), [email protected] (Y. Tauman). 0899-8256/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2004.10.006
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person from making the effort to mimic him. This is a credible signal to an uninformed employer about the ability of the individual. The high ability individual thus over-invests in education and certainly acquires more education than the one with low ability. However, occasionally we also observe the reverse phenomenon of very able individuals regarding investment in education as a waste of time. This willingness to cut education short is sometimes a credible signal of strength. The purpose of this paper is to study the phenomenon of some gifted entrepreneurs acquiring relatively low levels of education and in some extreme cases dropping out of schools shortly after enrolling.1 The paper considers two types of entrepreneurs: ordinary and gifted. In contrast to Spence (1974), in this paper the productivity of both the gifted and ordinary individuals increases with the level of education. In every separating equilibrium the ordinary individual acquires his myopic optimal level (the one that would be chosen in the case of complete information). As for the gifted individual we find circumstances where he acquires an education level that is sufficiently lower than that of the ordinary individual. The ordinary individual will not mimic the gifted individual since such a low level of education will significantly reduce his productivity. On the other hand, the gifted individual can afford a low level of education since his productivity is significantly higher than that of the ordinary individual, even for a low level of education. The individual in this paper wishes to become a successful entrepreneur. However he does not have sufficient economic resources to start his own business and depends on external funds. Thus, he has to convince potential investors to invest in the enterprise in return for a certain ownership share. The type of an individual (ordinary or gifted) is not known to potential investors. The expected net value of an enterprise run by an ordinary entrepreneur is negative and it is positive if it is run by a gifted entrepreneur. In our model the individual is active for one unit of time which is divided into two periods. The first is the education period and its length is a strategic variable of the individual. In the second period the individual can either establish his own enterprise (if he finds an investor to fund it) or approach a competitive job market. The type of the entrepreneur and his education level affect his probability to run a successful enterprise. It is assumed that this probability is higher for the gifted individual than for the ordinary one irrespective of their education levels. The ordinary individual benefits from higher education in the sense that his probability to run a successful enterprise is strictly increasing with the level of education. It is shown that if the value per dollar investment of a successful enterprise is sufficiently larger and if the investment level is significant, then there exists a sensible separating equilibrium outcome. Moreover, in every sensible separating equilibrium, the ordinary individual acquires his myopic education level and approaches the job market while the gifted individual acquires an education level which is lower than that of the ordinary one and starts his own business. If the education level has a significant impact 1 There are many examples of hugely successful entrepreneurs who dropped out of school, even among the
2001 Forbes Four Hundred (the 400 richest Americans in 2001). For the complete list of dropouts from the 1996 Forbes Four Hundred, see Orzach and Tauman (1996) and Orzach (1997). Among them, Bill Gates dropped out of Harvard, Lawrence Ellison dropped out of the University of Illinois, Leslie Wexner dropped out of Ohio State University, Michael Dell of Dell Computers dropped out of the University of Texas, and others.
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also on the probability of the gifted individual to run a successful enterprise, then his myopic education level is higher than that of the ordinary individual, but still in a separating equilibrium he curtails his education level at a level below the ordinary one. To reduce the strategic dropout phenomenon, sophisticated education programs should be offered which are ‘over the head’ of ordinary individuals and relatively easy for gifted individuals. Such programs will only slightly increase the productivity of ordinary individuals and will be costly for them. The signaling reversal phenomenon in education models has already been studied by Swinkels (1999), Feltovich et al. (2003) (hereafter FHT) and Hvide (2003). Swinkels (1999) considered a version of Spence’s job market signaling model and showed that when education positively affects productivity, less able individuals will become overeducated to partially pool with more able individuals. Consequently, able individuals reduce their education to a level which is still above that of the less able ones. Applying the general model of FHT to education, the countersignaling result is obtained where high able individuals demonstrate their strength by curtailing education to a level below that of medium able individuals. Their model however differs from the main stream education models. They assumed that the employer obtains in addition to the standard signal of the individual’s education level (which is determined endogenously), a noisy signal about the individual’s type which is exogenously determined (e.g. a recommendation letter from a previous employer). The combination of the two signals enables a high type individual to separate himself from the other types with a countersignal. We obtain a similar result without the use of a second noisy exogenous signal. Our setup is quite simple and the result is obtained in a standard job market signaling approach. Finally, Hvide (2003) considers a two-sector job market model where individuals do not know their own type. It is shown that an individual who strongly believes that he is of the high type does not acquire education and immediately approaches the sector which rewards performance. He then succeeds to separate himself from the one who strongly believes that he is the low type and hence approaches the sector which offers wage contracts which are not contingent on performance. Individuals who are uncertain with a significant probability about their type acquire education, then learn their type and approach a proper sector.
2. The model Consider an individual E a potential entrepreneur who is active for one unit of time. The unit interval [0, 1] describes the relevant time horizon of E. The individual can be one of two types: G (Gifted) or N (Non-gifted), referred to in the Introduction as “ordinary.” The type of E is private information. Denote by Et the individual E of type t, t ∈ {G, N}. The interval [0, 1] is divided into two periods. The first period of length x is the time that E decides to spend on education. The value of x is a strategic choice of E and it is assumed that it is bounded above by T , 0 < T < 1, namely, 0 x T . The value of x becomes commonly known in the end of the first period. We denote by Ct (x) the cost of Et of acquiring eduction for x units of time. In the second period, E decides whether to establish his own enterprise or to approach the job market for employment. If he chooses to start
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his own business he will approach investors to raise a fund. It is assumed that E needs M dollars to establish and to run the business (E has no resources of his own). The investors are acting in a competitive environment. A contract between E and an investor I specifies the ownership shares α and 1 − α of E and I respectively. The outcome of the enterprise can be either S (Successful) or F (Failure). If the outcome is S then the monetary value is V , and if it is F then the monetary value is zero. It is assumed that V does not depend on E’s type. On the other hand, the probability of success does depend on E’s type. Let Pt (x) be the probability that Et who studies x will succeed with his enterprise, t ∈ {G, N}. The investors do not know if E is gifted or not and x serves as a signal for E’s type. The job market is assumed to be competitive and the salary of E will depend on the belief of the employer about E’s type. In the full information case where E’s type is commonly known Et obtains his marginal productivity (in monetary terms) Wt (x) per unit of time. Again, the employer may use x as a signal to E’s ability and hence to determine E’s salary. We focus on the case where the investors have no incentive to invest in an enterprise which is run by a non-gifted individual (if they happen to know his type). Namely, max PN (x)V − M < 0.
0xT
(2.1)
While (2.1) asserts that the expected net value of an enterprise run by a non-gifted individual, EN , is negative, there is a positive probability that EN will run a successful enterprise. This follows from our Assumption 1, below, that PN (x) is strictly increasing in x. Without this assumption EN will have no incentive to pretend to be the gifted individual and EG will act in the same way as in the full information case. If EN acquires a significant level of education and obtains a significant share of the enterprise then he will be better off starting his business (this is established later on in Proposition 1).2 Suppose that PG (x)V − M > 0 . The competition among investors will force them to sign the contract (α(x) ,1 − α(x)) with E (if they believe that the individual is gifted), where 1 − α(x) PG (x)V = M. Equivalently E’s share in the enterprise is α(x) = 1 −
M PG (x)V
(2.2)
and he obtains PG (x)V − M. It is required that for all x, 0 x T , PG (x)V − M > (1 − x)WG (x).
(2.3)
2 The inequality (2.1) is not crucial for the strategic dropout phenomenon. If for some (large) values of x,
PN (x)V − M > 0 then for these values the investor may finance the enterprise (since the expected value of the enterprise is positive for both types of individual). If PN (x) is significantly smaller than PG (x) then in the full information case the investor will offer EG a significantly higher share than to EN . Therefore in the private information case (and for proper values of V and M) EG is best off signaling his type with a small x. The nongifted individual EN will find it not attractive to mimic EG since for sufficiently small x the share increase will not compensate for the small probability of success.
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The last inequality implies that (i) The investors will fund the enterprise if they believe that the individual is gifted. (ii) A gifted individual prefers to establish his own enterprise rather than to approach the job market even if the employer correctly identifies his type. We next introduce some payoff functions which play a crucial role in the sequel analysis. Let UNN (x) be the payoff of EN if his type is correctly identified by the investors and by the employers. In this case EN will not be able to fund his enterprise and he will approach the job market. His net payoff is then UNN (x) = (1 − x)WN (x) − CN (x).
(2.4)
Consider a non-gifted individual who acquires x and succeeds to fool the investors to believe that he is gifted. If he decided to start an enterprise, the investors will fund it, he will obtain the share α(x) and his payoff is UNG (x) = PN (x)V α(x) − CN (x). By (2.2),
UNG (x) = PN (x) V −
M − CN (x). PG (x)
(2.5)
(2.6)
The payoff UNG (x) applies only to the case where EN starts his own business whenever he can find an investor to fund it. We state now the assumptions of the model. Assumption 1. The probability PN (x) is strictly increasing and (weakly) concave in x, 0 x T , and PN (0) = 0. Furthermore, the probability PG (x) is nondecreasing and PG (0) > PN (T ). This means that the gifted individual has a higher probability to establish a successful enterprise irrespective of the education level of the non-gifted individual. Note that the assumption that PN (x) is strictly increasing, is different from the standard literature of the signaling education model. This literature treats education as a waste of effort which is used just as a costly (and therefore credible) signal for the ability of the individual. Here, higher education improves the probability of turning an enterprise into a successful business (in addition to the information it conveys about the individual’s type). Assumption 2. The wage Wt (x), t ∈ {G, N} is nondecreasing. WN (x) is (weakly) concave in x and WN (x) WG (x) for all x ∈ [0, T ]. Assumption 3. The cost function Ct (x) is strictly convex and Ct (0) = 0 for t ∈ {G, N}. Assumption 4. The functions Pt (x), Wt (x), and Ct (x) are continuously differentiable in [0, T ), t ∈ {G, N}.
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By Assumptions 2 and 3 it easily follows that UNN (x) is strictly concave and thus attains its maximum in a single point. Let m xN = arg max UNN (x). 0xT
(2.7)
(0) > 0. This implies that x m > 0. Assumption 5. UNN N
This is a technical but necessary assumption. Our purpose is to show that a non-gifted individual may acquire a higher education level than a gifted individual. Since in every m , we must separating equilibrium the non-gifted individual acquires the education level xN m have xN > 0. Furthermore, this separates non-gifted ordinary individuals from the noncompetent individuals who are not able to study higher education (or to be admitted to reasonable higher level schools). We did not include in our model the third type of noncompetent individuals since no insight would be gained. 2.1. Sensible sequential separating equilibria Let xt∗ be the equilibrium education level of Et , t ∈ {G, N}. We confine our analysis to ∗ = x ∗ . Also, we are interested in sequential pure strategy separating3 equilibrium where xG N equilibrium points where the gifted individuals start their own business. Finally we restrict our attention to ‘sensible’ equilibrium points, which satisfy the intuitive criterion of Cho and Kreps (1987). Let us denote these equilibrium points by SSSE (sensible, sequential ∗ is observed, the and separating equilibrium). Consider a separating equilibrium. When xN ∗ = x ∗ . Thus, E obtains investors and the employers infer that E is of type N since xN N G m ∗ ). This implies that x ∗ = x m since otherwise U ∗ UNN (xN NN (xN ) > UNN (xN ). On the other N N ∗ will offer E a contract (α( x ∗ ), 1 − α (x ∗ )). By (2.2), hand, an investor who observes xG G G ∗ ) = 1 − M/(P (x ∗ )V ). α(xG G G ∗ let X ⊆ [0, T ] be defined by To determine xG m UNG (x). (2.8) x ∈ X iff UNN xN Since UNG (x) is continuous in x, X is compact. Next define UGG (x) = PG (x)α(x)V − CG (x) = PG (x)V − M − CG (x).
(2.9)
The function UGG (x) is the payoff of EG if his type is correctly identified by the investors. Note that UGG (x) is continuous in x. Hence arg maxx∈X UGG (x) = φ. ∗ ∈ arg max Lemma 1. In every SSSE xG x∈X UGG (x). 3 Note that a necessary condition for a pooling equilibrium to exist is that the proportion, η, of gifted individuals
is sufficiently large. Namely, for some x1 and for x x1 , ηPG (x) + (1 − η)PN (x) V − M 0. Otherwise, investors will never invest in the enterprise. Our analysis certainly applies for cases where the investment is significant (relative to V ) and the proportion of gifted individuals is relatively small.
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Proof. See Appendix A.
85
2
∗ < x ∗ = x m. Our next goal is to find conditions which guarantee that xG N N
Proposition 1. Suppose that m ) C (T ) WN (xN M N + max V> m) m ) , P (T ) . PG (xN PN (xN N
(2.10)
m ) and therefore x ∗ < x m = x ∗ . Then X ⊆ [0, xN N G N
Proof. First notice that by (2.6) and Assumption 1, UNG (0) = 0 and hence UNG (0) < m m m UNN (xN ). Next let us prove that UNG (x) > UNN (xN ) for all x, xN x T . Denote NG (x) = PN (x) V − M U − CN (x), 0 x T . m PG (xN ) NG (x) UNG (x) for all x, x m x T . Clearly4 by (2.6) U N Observe that for all x, 0 x T , M UNG (x) = PN (x) V − m ) − CN (x). PG (xN By the concavity of PN (x) and the convexity of CN (x), M NG (x) > PN (T ) V − U m ) − CN (T ). PG (xN NG (x m ) (x) > 0 for all x. Consequently, if we prove that UNN (x m ) < U By (2.10), U N N NG m m then it will follow that UNN (xN ) < UNG (x) for all xN x T . But the inequalm) < U NG (x m ) follows immediately by (2.10) and (2.4). This, together with ity UNN (xN N m UNG (0) < UNN (xN ), implies that X = ∅ (as it contains a small neighborhood of 0) and m ). By Lemma 1, x ∗ ∈ X and since X ⊆ [0, x m ) we have that x ∗ < x ∗ . 2 X ⊆ [0, xN N G G N Next we examine the incentive compatible condition of the gifted individual. Denote by UGN (x) the payoff of a gifted individual if he is perceived by the investor as non-gifted when choosing x. That is, UGN (x) = (1 − x)W (x) − CG (x)
(2.11)
where WN (x) W (x) WG (x). Note that W (x) depends on the beliefs of the employer about the individual’s type. The incentive compatible constraint of the gifted individual is certainly satisfied5 if for all x ∗ UGG xG UGN (x). (2.12) 4 The function U NG (x) is just an auxiliary function and does not describe any payoff in this game. 5 Note that it is sufficient that U ∗ / int X (since for all x ∈ int X the investors in GG (xG ) UGN (x) whenever x ∈
every SSSE believe that the individual is gifted with probability 1 and hence they will fund him).
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∗ ∈ X and 0 ∈ X, by Corollary 1 it is sufficient that for all x Since xG
UGG (0) (1 − x)WG (x) − CG (x). By (2.9), the last inequality holds if PG (0)V − M WG (T ) or if M + WG (T ) . (2.13) PG (0) To complete the analysis we need to ensure that conditions (2.1) and (2.3) are satisfied. The first one is PN (x)V − M < 0 for all 0 x T , which is equivalent to V
M . (2.14) PN (T ) Finally, it is straightforward that (2.13) implies condition (2.3). Combining the conditions of Proposition 1 together with (2.13) and (2.14) we have V<
M M +A
m ) C (T ) WG (T ) WN (xN N A = max , m ) , P (T ) PG (0) PN (xN N
(2.15)
or equivalently, 1 A V 1 + < < . (2.16) PG (0) M M PN (T ) Since PG (0) > PN (T ) (Assumption 1), inequality (2.16) holds for V and M sufficiently large (provided V is not too large). We can summarize the above as follows: Theorem 2.1. Suppose that A V 1 1 + < < . PG (0) M M PN (T ) ∗ < x∗ = xm. Then there exists an SSSE and in every SSSE, 0 < xG N N
The existence of the equilibrium follows from the above discussion and it is supported for instance by the investors’ beliefs that the individual is gifted with probability 1 whenever x ∈ X and he is non-gifted whenever x ∈ / X. Next note that the inequality V /M < 1/PN (T ) is exactly (2.1). To satisfy the inequality 1/PG (0) + A/M < 1/PN (T ), M should be sufficiently large. This implies that V should also be large (to satisfy 1/PG (0) + A/M < V /M) but not too large to provide no incentive for the investor to invest in the non-gifted individual irrespective of his education level. Namely, to guarantee that V PN (T ) < M. Suppose next that M and V are fixed. Then the condition of the theorem requires that m ), PN (T ) and PG (0) should not be too small. This together with PN (0) = 0 imPN (xN ply the signal reversal phenomenon whenever a higher education program significantly improves the probability of a non-gifted individual to run a successful enterprise.
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Consequently, schools that offer just cookbook programs which are useful for non-gifted individuals will induce gifted individuals to dropout since they will not be able to separate themselves from non-gifted individuals by acquiring higher education. Finally, suppose that education has a significant impact also on the probability of the gifted individual to run a successful enterprise. Then with an appropriate PG (x) function, the myopic education level of the gifted individual could be made as close to T as we wish. In this case the result is even more striking. The myopic education level of the gifted individual is greater than that of a non-gifted individual, but the only way he can separate himself from the non-gifted individual is by curtailing his education below even the myopic level of the non-gifted individual.
Acknowledgments We gratefully acknowledge the comments and suggestions made by the editor Andrew Postlewaite and an anonymous referee.
Appendix A. Proof of Lemma 1 ∗ ∈ Suppose to the contrary that there exists an SSSE where xG / arg maxx∈X UGG (x). Let xmax ∈ arg maxx∈X UGG (x). Since UGG (x) is continuous in x and X is compact, there ∗ + (1 − β)x exists β ∈ (0, 1) such that for xB = βxG max the following holds: m (i) UNG (xB ) < UNN (xN ), and ∗ (ii) UGG (xB ) > UGG (xG ). ∗ is the equilibrium action of E , by (ii) it must be that E at x is offered a Since xG G B share α s.t. α < α(xB ). That is, at xB the investor I believes that E is gifted with probability µ < 1. Thus, I will offer E at xB a share M , 0 < α(xB ) αµ = max 1 − Vλ
where λ = µPG (xB ) + (1 − µ)PN (xB ). By the definition of λ and by (i), m UNλ (xB ) < UNG (xB ) < UNN (xN )
where UNλ (xB ) = PN (xB )V αµ − CN (xB ) is the expected payoff of EN when I believes at xB that EN is of type G with probability µ < 1 and hence will offer EN the share αµ . If αµ is sufficiently attractive for EN to start his own business, then UNλ (xB ) < UNG (xB ) m . By the intuitive (as αµ < α(xB )). By (i), xB is an inferior action for EN compared to xN criterion, if I observes xB he will assign probability 1 that E is of type G. But then I will offer E the share α(xB ), a contradiction. If EN at xB is better off approaching the job market (given αµ ), then again I when observing xB will offer E the share α(xB ) (knowing that ∗ ∈ arg max E is of type G), a contradiction. Consequently, in every SSSE xG x∈X UGG (x).
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References Cho, I.K., Kreps, D.M., 1987. Signaling games and stable equilibria. Quart. J. Econ. 102, 179–221. Feltovich, N., Harbaugh, R., To, T., 2003. Too cool for school? Signaling and countersignaling. RAND J. Econ. 33 (4), 630–649. Hvide, H., 2003. Education and the allocation of talent. J. Lab. Econ. 21 (4), 945–976. Orzach, R., Tauman, Y., 1996. Strategic dropouts. Working paper. Institute for Decision Sciences, SUNY at Stony Brook. Orzach, R., 1997. Signalling reversal. PhD thesis, Chapter 3. Tel Aviv University. Spence, M., 1974. Market Signalling. Harvard Univ. Press, Cambridge, MA. Swinkels, J., 1999. Education signalling with preemptive offers. Rev. Econ. Stud. 66, 949–970.
Strategic dropouts Ram Orzach a,b,∗ , Yair Tauman c,d a Wayne State University, USA b Tulane University, USA c Tel Aviv University, Israel d SUNY, Stony Brook, USA
Received 10 September 2003 Available online 8 December 2004
Abstract Following Spence, this note provides an education signaling model to explain the phenomenon of gifted entrepreneurs who acquire less education than ordinary individuals. Two types of individuals, ordinary and gifted, are considered. Each one of them can either convince an investor to fund his enterprise or approach a competitive job market. The probability that an ordinary individual succeeds to establish a successful enterprise is smaller than that of a gifted individual irrespective of his education level. The probability of an ordinary individual succeeding increases with the level of education. In a separating equilibrium gifted individuals curtail their eduction to a level below that of the ordinary ones. This happens if the value of a successful enterprise per dollar investment is sufficiently large, on the one hand, but not too large to guarantee that the expected value of an enterprise run by an educated ordinary entrepreneur falls below the investment cost, on the other. 2004 Elsevier Inc. All rights reserved. JEL classification: C72; D82
1. Introduction Spence’s (1974) signaling education model focuses on the phenomenon of a high-ability individual investing significantly in education to a level just sufficient to deter a less able * Corresponding author.
E-mail addresses: [email protected] (R. Orzach), [email protected] (Y. Tauman). 0899-8256/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.geb.2004.10.006
80
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person from making the effort to mimic him. This is a credible signal to an uninformed employer about the ability of the individual. The high ability individual thus over-invests in education and certainly acquires more education than the one with low ability. However, occasionally we also observe the reverse phenomenon of very able individuals regarding investment in education as a waste of time. This willingness to cut education short is sometimes a credible signal of strength. The purpose of this paper is to study the phenomenon of some gifted entrepreneurs acquiring relatively low levels of education and in some extreme cases dropping out of schools shortly after enrolling.1 The paper considers two types of entrepreneurs: ordinary and gifted. In contrast to Spence (1974), in this paper the productivity of both the gifted and ordinary individuals increases with the level of education. In every separating equilibrium the ordinary individual acquires his myopic optimal level (the one that would be chosen in the case of complete information). As for the gifted individual we find circumstances where he acquires an education level that is sufficiently lower than that of the ordinary individual. The ordinary individual will not mimic the gifted individual since such a low level of education will significantly reduce his productivity. On the other hand, the gifted individual can afford a low level of education since his productivity is significantly higher than that of the ordinary individual, even for a low level of education. The individual in this paper wishes to become a successful entrepreneur. However he does not have sufficient economic resources to start his own business and depends on external funds. Thus, he has to convince potential investors to invest in the enterprise in return for a certain ownership share. The type of an individual (ordinary or gifted) is not known to potential investors. The expected net value of an enterprise run by an ordinary entrepreneur is negative and it is positive if it is run by a gifted entrepreneur. In our model the individual is active for one unit of time which is divided into two periods. The first is the education period and its length is a strategic variable of the individual. In the second period the individual can either establish his own enterprise (if he finds an investor to fund it) or approach a competitive job market. The type of the entrepreneur and his education level affect his probability to run a successful enterprise. It is assumed that this probability is higher for the gifted individual than for the ordinary one irrespective of their education levels. The ordinary individual benefits from higher education in the sense that his probability to run a successful enterprise is strictly increasing with the level of education. It is shown that if the value per dollar investment of a successful enterprise is sufficiently larger and if the investment level is significant, then there exists a sensible separating equilibrium outcome. Moreover, in every sensible separating equilibrium, the ordinary individual acquires his myopic education level and approaches the job market while the gifted individual acquires an education level which is lower than that of the ordinary one and starts his own business. If the education level has a significant impact 1 There are many examples of hugely successful entrepreneurs who dropped out of school, even among the
2001 Forbes Four Hundred (the 400 richest Americans in 2001). For the complete list of dropouts from the 1996 Forbes Four Hundred, see Orzach and Tauman (1996) and Orzach (1997). Among them, Bill Gates dropped out of Harvard, Lawrence Ellison dropped out of the University of Illinois, Leslie Wexner dropped out of Ohio State University, Michael Dell of Dell Computers dropped out of the University of Texas, and others.
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also on the probability of the gifted individual to run a successful enterprise, then his myopic education level is higher than that of the ordinary individual, but still in a separating equilibrium he curtails his education level at a level below the ordinary one. To reduce the strategic dropout phenomenon, sophisticated education programs should be offered which are ‘over the head’ of ordinary individuals and relatively easy for gifted individuals. Such programs will only slightly increase the productivity of ordinary individuals and will be costly for them. The signaling reversal phenomenon in education models has already been studied by Swinkels (1999), Feltovich et al. (2003) (hereafter FHT) and Hvide (2003). Swinkels (1999) considered a version of Spence’s job market signaling model and showed that when education positively affects productivity, less able individuals will become overeducated to partially pool with more able individuals. Consequently, able individuals reduce their education to a level which is still above that of the less able ones. Applying the general model of FHT to education, the countersignaling result is obtained where high able individuals demonstrate their strength by curtailing education to a level below that of medium able individuals. Their model however differs from the main stream education models. They assumed that the employer obtains in addition to the standard signal of the individual’s education level (which is determined endogenously), a noisy signal about the individual’s type which is exogenously determined (e.g. a recommendation letter from a previous employer). The combination of the two signals enables a high type individual to separate himself from the other types with a countersignal. We obtain a similar result without the use of a second noisy exogenous signal. Our setup is quite simple and the result is obtained in a standard job market signaling approach. Finally, Hvide (2003) considers a two-sector job market model where individuals do not know their own type. It is shown that an individual who strongly believes that he is of the high type does not acquire education and immediately approaches the sector which rewards performance. He then succeeds to separate himself from the one who strongly believes that he is the low type and hence approaches the sector which offers wage contracts which are not contingent on performance. Individuals who are uncertain with a significant probability about their type acquire education, then learn their type and approach a proper sector.
2. The model Consider an individual E a potential entrepreneur who is active for one unit of time. The unit interval [0, 1] describes the relevant time horizon of E. The individual can be one of two types: G (Gifted) or N (Non-gifted), referred to in the Introduction as “ordinary.” The type of E is private information. Denote by Et the individual E of type t, t ∈ {G, N}. The interval [0, 1] is divided into two periods. The first period of length x is the time that E decides to spend on education. The value of x is a strategic choice of E and it is assumed that it is bounded above by T , 0 < T < 1, namely, 0 x T . The value of x becomes commonly known in the end of the first period. We denote by Ct (x) the cost of Et of acquiring eduction for x units of time. In the second period, E decides whether to establish his own enterprise or to approach the job market for employment. If he chooses to start
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his own business he will approach investors to raise a fund. It is assumed that E needs M dollars to establish and to run the business (E has no resources of his own). The investors are acting in a competitive environment. A contract between E and an investor I specifies the ownership shares α and 1 − α of E and I respectively. The outcome of the enterprise can be either S (Successful) or F (Failure). If the outcome is S then the monetary value is V , and if it is F then the monetary value is zero. It is assumed that V does not depend on E’s type. On the other hand, the probability of success does depend on E’s type. Let Pt (x) be the probability that Et who studies x will succeed with his enterprise, t ∈ {G, N}. The investors do not know if E is gifted or not and x serves as a signal for E’s type. The job market is assumed to be competitive and the salary of E will depend on the belief of the employer about E’s type. In the full information case where E’s type is commonly known Et obtains his marginal productivity (in monetary terms) Wt (x) per unit of time. Again, the employer may use x as a signal to E’s ability and hence to determine E’s salary. We focus on the case where the investors have no incentive to invest in an enterprise which is run by a non-gifted individual (if they happen to know his type). Namely, max PN (x)V − M < 0.
0xT
(2.1)
While (2.1) asserts that the expected net value of an enterprise run by a non-gifted individual, EN , is negative, there is a positive probability that EN will run a successful enterprise. This follows from our Assumption 1, below, that PN (x) is strictly increasing in x. Without this assumption EN will have no incentive to pretend to be the gifted individual and EG will act in the same way as in the full information case. If EN acquires a significant level of education and obtains a significant share of the enterprise then he will be better off starting his business (this is established later on in Proposition 1).2 Suppose that PG (x)V − M > 0 . The competition among investors will force them to sign the contract (α(x) ,1 − α(x)) with E (if they believe that the individual is gifted), where 1 − α(x) PG (x)V = M. Equivalently E’s share in the enterprise is α(x) = 1 −
M PG (x)V
(2.2)
and he obtains PG (x)V − M. It is required that for all x, 0 x T , PG (x)V − M > (1 − x)WG (x).
(2.3)
2 The inequality (2.1) is not crucial for the strategic dropout phenomenon. If for some (large) values of x,
PN (x)V − M > 0 then for these values the investor may finance the enterprise (since the expected value of the enterprise is positive for both types of individual). If PN (x) is significantly smaller than PG (x) then in the full information case the investor will offer EG a significantly higher share than to EN . Therefore in the private information case (and for proper values of V and M) EG is best off signaling his type with a small x. The nongifted individual EN will find it not attractive to mimic EG since for sufficiently small x the share increase will not compensate for the small probability of success.
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The last inequality implies that (i) The investors will fund the enterprise if they believe that the individual is gifted. (ii) A gifted individual prefers to establish his own enterprise rather than to approach the job market even if the employer correctly identifies his type. We next introduce some payoff functions which play a crucial role in the sequel analysis. Let UNN (x) be the payoff of EN if his type is correctly identified by the investors and by the employers. In this case EN will not be able to fund his enterprise and he will approach the job market. His net payoff is then UNN (x) = (1 − x)WN (x) − CN (x).
(2.4)
Consider a non-gifted individual who acquires x and succeeds to fool the investors to believe that he is gifted. If he decided to start an enterprise, the investors will fund it, he will obtain the share α(x) and his payoff is UNG (x) = PN (x)V α(x) − CN (x). By (2.2),
UNG (x) = PN (x) V −
M − CN (x). PG (x)
(2.5)
(2.6)
The payoff UNG (x) applies only to the case where EN starts his own business whenever he can find an investor to fund it. We state now the assumptions of the model. Assumption 1. The probability PN (x) is strictly increasing and (weakly) concave in x, 0 x T , and PN (0) = 0. Furthermore, the probability PG (x) is nondecreasing and PG (0) > PN (T ). This means that the gifted individual has a higher probability to establish a successful enterprise irrespective of the education level of the non-gifted individual. Note that the assumption that PN (x) is strictly increasing, is different from the standard literature of the signaling education model. This literature treats education as a waste of effort which is used just as a costly (and therefore credible) signal for the ability of the individual. Here, higher education improves the probability of turning an enterprise into a successful business (in addition to the information it conveys about the individual’s type). Assumption 2. The wage Wt (x), t ∈ {G, N} is nondecreasing. WN (x) is (weakly) concave in x and WN (x) WG (x) for all x ∈ [0, T ]. Assumption 3. The cost function Ct (x) is strictly convex and Ct (0) = 0 for t ∈ {G, N}. Assumption 4. The functions Pt (x), Wt (x), and Ct (x) are continuously differentiable in [0, T ), t ∈ {G, N}.
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By Assumptions 2 and 3 it easily follows that UNN (x) is strictly concave and thus attains its maximum in a single point. Let m xN = arg max UNN (x). 0xT
(2.7)
(0) > 0. This implies that x m > 0. Assumption 5. UNN N
This is a technical but necessary assumption. Our purpose is to show that a non-gifted individual may acquire a higher education level than a gifted individual. Since in every m , we must separating equilibrium the non-gifted individual acquires the education level xN m have xN > 0. Furthermore, this separates non-gifted ordinary individuals from the noncompetent individuals who are not able to study higher education (or to be admitted to reasonable higher level schools). We did not include in our model the third type of noncompetent individuals since no insight would be gained. 2.1. Sensible sequential separating equilibria Let xt∗ be the equilibrium education level of Et , t ∈ {G, N}. We confine our analysis to ∗ = x ∗ . Also, we are interested in sequential pure strategy separating3 equilibrium where xG N equilibrium points where the gifted individuals start their own business. Finally we restrict our attention to ‘sensible’ equilibrium points, which satisfy the intuitive criterion of Cho and Kreps (1987). Let us denote these equilibrium points by SSSE (sensible, sequential ∗ is observed, the and separating equilibrium). Consider a separating equilibrium. When xN ∗ = x ∗ . Thus, E obtains investors and the employers infer that E is of type N since xN N G m ∗ ). This implies that x ∗ = x m since otherwise U ∗ UNN (xN NN (xN ) > UNN (xN ). On the other N N ∗ will offer E a contract (α( x ∗ ), 1 − α (x ∗ )). By (2.2), hand, an investor who observes xG G G ∗ ) = 1 − M/(P (x ∗ )V ). α(xG G G ∗ let X ⊆ [0, T ] be defined by To determine xG m UNG (x). (2.8) x ∈ X iff UNN xN Since UNG (x) is continuous in x, X is compact. Next define UGG (x) = PG (x)α(x)V − CG (x) = PG (x)V − M − CG (x).
(2.9)
The function UGG (x) is the payoff of EG if his type is correctly identified by the investors. Note that UGG (x) is continuous in x. Hence arg maxx∈X UGG (x) = φ. ∗ ∈ arg max Lemma 1. In every SSSE xG x∈X UGG (x). 3 Note that a necessary condition for a pooling equilibrium to exist is that the proportion, η, of gifted individuals
is sufficiently large. Namely, for some x1 and for x x1 , ηPG (x) + (1 − η)PN (x) V − M 0. Otherwise, investors will never invest in the enterprise. Our analysis certainly applies for cases where the investment is significant (relative to V ) and the proportion of gifted individuals is relatively small.
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Proof. See Appendix A.
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2
∗ < x ∗ = x m. Our next goal is to find conditions which guarantee that xG N N
Proposition 1. Suppose that m ) C (T ) WN (xN M N + max V> m) m ) , P (T ) . PG (xN PN (xN N
(2.10)
m ) and therefore x ∗ < x m = x ∗ . Then X ⊆ [0, xN N G N
Proof. First notice that by (2.6) and Assumption 1, UNG (0) = 0 and hence UNG (0) < m m m UNN (xN ). Next let us prove that UNG (x) > UNN (xN ) for all x, xN x T . Denote NG (x) = PN (x) V − M U − CN (x), 0 x T . m PG (xN ) NG (x) UNG (x) for all x, x m x T . Clearly4 by (2.6) U N Observe that for all x, 0 x T , M UNG (x) = PN (x) V − m ) − CN (x). PG (xN By the concavity of PN (x) and the convexity of CN (x), M NG (x) > PN (T ) V − U m ) − CN (T ). PG (xN NG (x m ) (x) > 0 for all x. Consequently, if we prove that UNN (x m ) < U By (2.10), U N N NG m m then it will follow that UNN (xN ) < UNG (x) for all xN x T . But the inequalm) < U NG (x m ) follows immediately by (2.10) and (2.4). This, together with ity UNN (xN N m UNG (0) < UNN (xN ), implies that X = ∅ (as it contains a small neighborhood of 0) and m ). By Lemma 1, x ∗ ∈ X and since X ⊆ [0, x m ) we have that x ∗ < x ∗ . 2 X ⊆ [0, xN N G G N Next we examine the incentive compatible condition of the gifted individual. Denote by UGN (x) the payoff of a gifted individual if he is perceived by the investor as non-gifted when choosing x. That is, UGN (x) = (1 − x)W (x) − CG (x)
(2.11)
where WN (x) W (x) WG (x). Note that W (x) depends on the beliefs of the employer about the individual’s type. The incentive compatible constraint of the gifted individual is certainly satisfied5 if for all x ∗ UGG xG UGN (x). (2.12) 4 The function U NG (x) is just an auxiliary function and does not describe any payoff in this game. 5 Note that it is sufficient that U ∗ / int X (since for all x ∈ int X the investors in GG (xG ) UGN (x) whenever x ∈
every SSSE believe that the individual is gifted with probability 1 and hence they will fund him).
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∗ ∈ X and 0 ∈ X, by Corollary 1 it is sufficient that for all x Since xG
UGG (0) (1 − x)WG (x) − CG (x). By (2.9), the last inequality holds if PG (0)V − M WG (T ) or if M + WG (T ) . (2.13) PG (0) To complete the analysis we need to ensure that conditions (2.1) and (2.3) are satisfied. The first one is PN (x)V − M < 0 for all 0 x T , which is equivalent to V
M . (2.14) PN (T ) Finally, it is straightforward that (2.13) implies condition (2.3). Combining the conditions of Proposition 1 together with (2.13) and (2.14) we have V<
M M +A
m ) C (T ) WG (T ) WN (xN N A = max , m ) , P (T ) PG (0) PN (xN N
(2.15)
or equivalently, 1 A V 1 + < < . (2.16) PG (0) M M PN (T ) Since PG (0) > PN (T ) (Assumption 1), inequality (2.16) holds for V and M sufficiently large (provided V is not too large). We can summarize the above as follows: Theorem 2.1. Suppose that A V 1 1 + < < . PG (0) M M PN (T ) ∗ < x∗ = xm. Then there exists an SSSE and in every SSSE, 0 < xG N N
The existence of the equilibrium follows from the above discussion and it is supported for instance by the investors’ beliefs that the individual is gifted with probability 1 whenever x ∈ X and he is non-gifted whenever x ∈ / X. Next note that the inequality V /M < 1/PN (T ) is exactly (2.1). To satisfy the inequality 1/PG (0) + A/M < 1/PN (T ), M should be sufficiently large. This implies that V should also be large (to satisfy 1/PG (0) + A/M < V /M) but not too large to provide no incentive for the investor to invest in the non-gifted individual irrespective of his education level. Namely, to guarantee that V PN (T ) < M. Suppose next that M and V are fixed. Then the condition of the theorem requires that m ), PN (T ) and PG (0) should not be too small. This together with PN (0) = 0 imPN (xN ply the signal reversal phenomenon whenever a higher education program significantly improves the probability of a non-gifted individual to run a successful enterprise.
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Consequently, schools that offer just cookbook programs which are useful for non-gifted individuals will induce gifted individuals to dropout since they will not be able to separate themselves from non-gifted individuals by acquiring higher education. Finally, suppose that education has a significant impact also on the probability of the gifted individual to run a successful enterprise. Then with an appropriate PG (x) function, the myopic education level of the gifted individual could be made as close to T as we wish. In this case the result is even more striking. The myopic education level of the gifted individual is greater than that of a non-gifted individual, but the only way he can separate himself from the non-gifted individual is by curtailing his education below even the myopic level of the non-gifted individual.
Acknowledgments We gratefully acknowledge the comments and suggestions made by the editor Andrew Postlewaite and an anonymous referee.
Appendix A. Proof of Lemma 1 ∗ ∈ Suppose to the contrary that there exists an SSSE where xG / arg maxx∈X UGG (x). Let xmax ∈ arg maxx∈X UGG (x). Since UGG (x) is continuous in x and X is compact, there ∗ + (1 − β)x exists β ∈ (0, 1) such that for xB = βxG max the following holds: m (i) UNG (xB ) < UNN (xN ), and ∗ (ii) UGG (xB ) > UGG (xG ). ∗ is the equilibrium action of E , by (ii) it must be that E at x is offered a Since xG G B share α s.t. α < α(xB ). That is, at xB the investor I believes that E is gifted with probability µ < 1. Thus, I will offer E at xB a share M , 0 < α(xB ) αµ = max 1 − Vλ
where λ = µPG (xB ) + (1 − µ)PN (xB ). By the definition of λ and by (i), m UNλ (xB ) < UNG (xB ) < UNN (xN )
where UNλ (xB ) = PN (xB )V αµ − CN (xB ) is the expected payoff of EN when I believes at xB that EN is of type G with probability µ < 1 and hence will offer EN the share αµ . If αµ is sufficiently attractive for EN to start his own business, then UNλ (xB ) < UNG (xB ) m . By the intuitive (as αµ < α(xB )). By (i), xB is an inferior action for EN compared to xN criterion, if I observes xB he will assign probability 1 that E is of type G. But then I will offer E the share α(xB ), a contradiction. If EN at xB is better off approaching the job market (given αµ ), then again I when observing xB will offer E the share α(xB ) (knowing that ∗ ∈ arg max E is of type G), a contradiction. Consequently, in every SSSE xG x∈X UGG (x).
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