College Enrollment, Dropouts and Option Value of Education

Transcription

1 College Enollment, Dopouts and Option Value of Education Ozdagli, Ali Tachte, Nicholas y Febuay 5, 2008 Abstact Psychic costs ae the most impotant component of the papes that ae tying to match empiical facts elated with college enollment and dopout decisions of high school gaduates. The absense of psychic cost equies vey high levels of isk avesion to duplicate the same facts. Howeve, psychic costs lack any economic meaning and teated as an obscue idiosyncatic component of utility function in in uential papes such as Keane and Wolpin (997) and Caneio, Hansen, Heckman (2003). In this pape, we model college enollment and dopout decisions in a eal options model simila to Miao and Wang (2007) whee the psychic cost stems fom the initial belief of students about thei skill level and the option value stems fom the Bayesian leaning pocess of one s skill level. Fist, ou model shows that papes that assume isk neutality oveestimate the value of college, college enollment and gaduation ates because the outcome of college education is isky. Second, we show that the option value of leaning is much moe impotant when agents ae isk avese athe than isk neutal. Finally, we suggest a welfae impoving policy that deceases both the pyschic costs, college enollment and gaduation ates wheeas a decease in the psychic costs in the afoementioned papes would incease both the college enollment and gaduation ates. Psychic costs ae the most impotant component of the papes that ae tying to match empiical facts elated with college enollment and dopout decisions of high school gaduates. The absense of psychic cost equies vey Ali and Nico ae fom The Univesity of Chicago. Contact: ozdagli@uchicago.edu y We wish to thank Fenando Alvaez, Las Hansen, James Heckman, Fabian Lange and Jada Boovicka fo vey helpful comments and suggestions. We also want to thank paticipants of the Macoeconomics Dynamics Woking Goup and Education Woking Goup at The Univesity of Chicago. All eos ae ou own.

2 high levels of isk avesion to duplicate the same facts. Howeve, psychic costs lack any economic meaning and ae teated as an obscue unobsevable idiosyncatic component of utility function in in uential papes such as Keane and Wolpin (997) and Caneio, Hansen, Heckman (2003). In this pape, we focus on college enollment and dopout decisions in a taceable options model simila to Miao and Wang (2007) whee the psychic cost stems fom the initial belief of students about thei skill level. The students with vey pessimistic initial beliefs do not enoll in college. If a student chooses to enoll in college he initial belief is updated duing the college education accoding to a Bayesian leaning pocess that may eventually lead to dopout decision. Accoding to ou model college education has two main bene ts. The st one is the sheepskin o cedential e ect because many of the high-skill jobs equie a college degee. The amount of time spent in college does not add anything to the students eaning potential if they do not nish the college with a degee. The second bene t comes fom leaning and the option to dop out of college. College students can choose to dop out of college if thei beliefs become so pessimistic duing the leaning pocess that decide to cut thei losses. We also assume that the students do not face any boowing constaints because of the availability of college loans. Cameon and Heckman (200), Keane and Wolpin (200), Cunha and Heckman (2002) and Cameon and Tabel (2004) povide empiical suppot fo this assumption. Ou pape has thee main esults. Fist, ou model shows that papes that assume isk neutality, such as Keane and Wolpin (997), Cunha and Heckman (2007) and Heckman and Navao (2007), oveestimate the value of college, college enollment and gaduation ates because the outcome of college education is isky. Second, we show that the option value of leaning is much moe impotant when agents ae isk avese athe than isk neutal. These two esults imply that any seious model of college education should deviate fom isk neutality assumption and any model with isk avese students should take the option value of dopping out into account. Finally, we suggest a welfae impoving policy that deceases both the pyschic costs, college enollment and gaduation ates wheeas a decease in the psychic costs in the afoementioned papes would incease both the college enollment and gaduation ates. We can intepet the isk-avesion also as ambiguity avesion because students assign subjective beliefs to di eent outcomes of schooling. Hansen and Sagent (200) show that the implications of isk-avesion and ambiguity avesion ae simila. 2

3 Model Envionment. We stat descibing the time zeo poblem of high school gaduates. The high school gaduates di e in thei types, in paticula, they can be skilled wokes (S) o unskilled wokes (U). A skilled student eans w S upon gaduation fom college wheeas an unskilled student o a college dopout eans w U < w S. Although a high school gaduate, indexed by j, does not know he type she has an initial belief about he pobability of being an unskilled woke which we denote as p j 0. Duing he college education she eceives good news at ate that tells he that she belongs to type S, that is p (t) 0 once good news ae eceived. Theefoe, a student that does not eceive a shock at date t updates he belief using Bayes ule. This gives us the evalution of p (t), the belief about the pobability of being type U: 2 dp (t) dt p (t) ( p (t)) with p (0) p j 0 Note that p 0 (t) > 0 because not eceiving any good news makes the agent moe pessimistic. The high school gaduate is endowed with initial wealth x j 0. The high school gaduates choose thei consumption steam, fc t : t 0g, and whethe to enoll in o dopout fom college given thei initial beliefs and wealth in ode to maximize thei time-sepaable expected discounted utility fom consumption. Z E 0 e t u (c t ) dt 2 0 p (t + dt) p (t + dt) p (t) p (t + dt) p (t) p (t + dt) p (t) p (t + dt) p (t) dt p (t) p (t) + ( p (t)) ( dt) p (t) p (t) + ( p (t)) ( dt) p (t) p (t) ( p (t) ( p (t)) ( dt)) p (t) + ( p (t)) ( dt) p (t) ( p (t)) dt ( p (t)) dt p (t) ( p (t)) ( p (t)) dt p(t+dt) p(t) Now, lim dt!0 p 0 (t) p (t) ( p (t)) dt 3

4 The discount ate > 0. The agents can boow and lend at a isk-fee inteest ate 0 <. A woke that eans wage w i accumulates wealth accoding to dx t dt x t + w i c t ; i 2 fu; Sg whee > 0 denotes the net inteest ate on assets. A high-school gaduate faces the decision of attending college, with costs of a pe unit of time, o joining the wok-foce diectly as an unskilled woke. It follows that the wealth dynamics fo a college student ae given by, dx t dt x t c t a We assume that the gaduation times fo the students that do not dop out ae exponentially distibuted. In paticula, the college students eceive a shock that makes them gaduate at ate. Futhe, we assume that if a student eceives a college diploma (i.e. gets hit by the shock) at date t she goes to the maket with the belief that she will wok in secto U with pobability p (t) and the teminal payo will be the coesponding expected continuation value given p (t). Then he tue type is fully evealed 3. Theefoe, it is possible fo a college gaduate to wok in the unskilled secto and this contibutes to the obseved wage dispesion of college gaduates that is much geate than the wage dispesion of highschool gaduates. As a esult, thee ae two souces of welfae loss in this model. Fist, the skilled wokes can be misallocated to unskilled secto eithe because thei initial beliefs ae bad o because they wee unlucky and did not eceive the good news befoe it is too late. Second, the unskilled wokes who ae optimistic about themselves would go to college although it is a waste of esouces because, as stated peviously, they will not be able to wok in the skilled secto.. Solving the model e c We will use u (c), whee > 0 denotes the coe cient of isk avesion. This utility function will geneate that wealth has no e ect on schooling decisions and thus simpli cates consideably all the calculations. 3 Hee we assume that an application to a skilled job pefectly eveal a student s tue type but that a student cannot apply fo a skilled job without having a college degee. This is also common in pactice as many ads and postings fo skilled jobs ask fo college degee explicitely. 4

5 Fistly we will povide the solution to the poblem of a woke of type i 2 fu; Sg. Then we will use this solution to solve the poblem of a student that aleady knows he is type S. Finally, with solutions to the pevious poblem we can tackle the poblem of a student that is uncetain about his tue type. An impotant undelying assumption in all of the following sections is that the suppot of x is the whole eal line (i.e. x 2 R). This assumption ties to captue the fact that student loans o incentives ae fully accesible to any potential college student... Woke of type i Let V (x; w i ) be the value fo an agent of woking in secto i and wealth x. The poblem faced by this agent can be witten as, e c V (x; w i ) max + V x (x; w i ) dx () c dt whee dx dt x + w i Poposition (Value at Wok) The solution to () is given by povided > > 0. Poof. See appendix. V (x; w i ) c (x; w i ) x + w i + c e x+w i + Note that, given w S > w U, V (x; w S ) > V (x; w U ) and so a woke of type S enjoys a highe level of welfae than a woke of type U. A natual extension of this poblem would be letting the w S and w U to be andomly dawn fom a given distibution in ode to analyze the e ects of the wage dispesion among college gaduates and high school gaduates. Fo a simple example 4, suppose that both of the wages ae nomally distibuted with mean w i and standad deviation i with S > U. Then the esults of ou model ae peseved once we let w i w i 2 2 i. 4 The nomal distibution of wages implies that the wages can be negative with some positive pobability. Not withstanding this shotcoming we will poceed with this example due to its simplicity. 5

6 ..2 Student of known type S Let W S (x) be the value of being a student with known type S and wealth x. The HJB equation fo this poblem can be witten as W S (x) max u (c t ) + V (x; w S ) W S (x) + W S dx c x t dt whee dx x c t a dt This equation is telling us that the ow value of being a student with wealth x accounts fo the instant utility, the change of value of switching to be a woke (with the coespondent pobability of this happening), and fo the maginal value of the change in wealth. Note that in (2) the dop-out option is not consideed. This comes fom the fact that povided the pio p 0 > 0 the agent was choosing to go to college, then with p 0 0 he would cetainly go to college, povided no boowing constaints ae pesent. Poposition 2 (Value of an S type student) The solution to (2) is given by W S (x) B e x+w S + c S x a + B (2) (3) whee B solves + ln ~ B (w S + a) ~B (4) povided > > 0. Poof. See appendix. Remak B exists, it is unique and it is the case that B 2 max ; e ((w S+a) ) ; + In the appendix we show the details about the popeties of B. 6

7 Lemma W S (x) is sticly inceasing and stictly concave. Poof. povided B > S 2 W S 2 Be..3 Student with unknown type x+w S + > 0 x+w S + < 0 Afte doing the petinent appoximation fom discete time, W P (x; p) max u (c) + ( p) W S (x) W p (x; p) + [pv (x; w U ) + ( p) V (x; w S ) W p (x; p)] (5) +Wx P dx dt + W p P dp (6) dt dx whee dt x c a p ( p) dp dt Poposition 3 (Value of an unknown student) The solution to (5) is given by W P (x; p) e (x+f(p)) c x + f (p) whee f (p) is the solution to the di eential equation + + ( p) + f 0 (p) (p ( p)) (f (p) + a) e f(p) ( p) Be w S + + pe w U + + ( p) e w S + (7) Poof. See appendix. Conjectue Thee exists p such that a student with belief p 2 [p ; ] dop-out of school. 7

8 Unde this conjectue we can ewite the value function of a student as 8 < W S (x) fo p 0 W (x; p) W P (x; p) fo p 2 (0; p ) : V (x; w U ) fo p 2 [p ; ] Note that thee is an undelying conjectue in constucting W (x; p). The conjectue is that W S (x) W P (x; p) V (x; w U ) fo the coesponding pios. The fact that such a p exists implies the existence of an optimal baie. Futhe, it must be the case that the maginal student with type p must be indi eent between staying at school and becoming a dop-out, W P (x; p ) V (x; w U ) (8) This condition is known as Value Matching (VM fom now on). Note that (8) implies that f (p + ) w U (9) Also, it must happen that thee is no exta value of staying in school fo this maginal student, W P p (x; p ) 0 (0) This condition is known as Smooth Pasting 5 (SP fom now on) and implies that it must be the case that f 0 (p ) 0 () The following lemma chaacteize uppe and lowe limits fo W P (x; p). Lemma 2 V (x; w U ) W P (x; p) < V (x; w S ) fo p < p. Poof. V (x; w U ) < W P (x; p) follows fom the fact that an agent can always dop out of school and get value V (x; w U ), and so if he emains a student it must have highe value. Then, if he is studying (i.e. p < p ), V (x; w U ) < W P (x; p). If he is studying is because the agent wants to become an S type woke and so 6. 5 this condition can be deived by setting up the poblem in discete time with initial pio p p and then appoximate into continuous time. 6 Povided a > 0 it can t be the case that V (x; w S) W P (x; p) even fo an agent that knows he is type S. 8

9 Lemma 3 w U + f (p) < w S + and f (p) positive and deceasing fo p 2 [0; p ]. Poof. Fom the pevious lemma we have that V (x; w U ) W P (x; p) < V (x; w S ) substituting into this condition the functional foms fo V (x; w U ), V (x; w S ) and W P (x; p) we obtain w U + f (p) < w S + ; 0 p p Futhe, given, f (p) > 0. To pove that f 0 (p) < 0 let s conjectue st that W S (x) > W p (x; p) because the student who is awae that he is skilled has to be stictly bette o (povided that > 0) than a student who is not sue about it. Fom this we nd that > Be w S + f(p) (2) Moeove, note that V (x; w U ) < W P (x; p) < V (x; w S ) fo p 2 (0; p ) implies w U + < f (p) < w S + We can ewite the di eential equation that f (p) has to satisfy as f 0 (p) (p ( p)) (p; f (p)) whee (p; f (p)) ( p) Be + pe w U + w S + ( ) + (f (p) + a) f(p) f(p) + ( p) e w S + f(p) (p) ( p) Be +pe w U + w S + f(p) f(p) + ( p) e w S + f(p) + 9

10 > 0. Futhe, Which implies + e w U + w S + f(p) f(p) e w S + > 0 because w S > w U and povided that (2) needs to hold. Now suppose that thee exist a value of p p < p such that f 0 (p ) 0. Theefoe, (p ; f (p )) > 0. Then thee exist a p 2 > p in a su ciently close neighbohood of p so that f (p 2 ) f (p ). (p 2 ; f (p 2 )) > (p ; f (p )) > Then, we should have > 0. Theefoe, > f 0 (p 2 ) > 0. If we would epeat this pocedue fo p 2 and futhe we can easily gue that if f (p) is inceasing fo a value of p p < p then it should be inceasing fo all p 2 (p ; p ). Given f (p) is a continuous function this implies that f (p ) > f (p ) > w U + contadicting the bounday condition that f (p ) w U +. As a esult, f 0 (p) < 0 fo all p < p. Povided VM and SP we can solve fo the optimal theshold p. Applying (9) and () to (7) and then solving fo p, p ( + (w U + a)) e (w S w U ) B ( + ) e (w S w U ) B Futhe, p 2 (0; ) povided ( + (w U + a)) e (w S w U ) B > 0 (3) o, ( + (w U + a)) e (w S w U ) > B (4) This two conditions will pove to be useful late. Now we can do some @w S > U @?? Although some of the sign of these patial deivatives ae vey intuitive some othes deseve special attention. We povide intuiton fo these deivatives. 0

11 : Think of schooling as a poject that causes utility loss today and utility gain tomoow. If inteest ate inceases both the discounted value of utility loss and utility gain will decease. Howeve, the decease in the discounted value of utility gain will be highe since it occus late in the futue. (To see this moe clealy think of a poject that equies investment A at the beginning fo T peiods and then will povide etuns B afte that and have positive discounted pesent value. The discounted value of this poject is A+(B+A)(+) T > 0. Taking the deivative shows that this value is deceasing in.) As a esult an incease in deceases the option value of schooling. This leads to a decease in p. > 0: If is highe students eceive the good news at a highe ate meaning that they get pessimistic faste. This would suggest that p should be lowe because not eceiving good news is moe infomative in a negative way. Howeve, highe also means that skilled students eceive the jump to p (t) 0 and to the highe continuation value W S (x) faste. In ou model, this latte e ect dominates the fome one indicating that p should incease has an ambiguous sign. Fo example, fo given paamete values in the benchmak case in the next section we have dp d > 0 wheeas fo w U 0:04 we have dp d < 0. The eason is that inceased isk avesion deceases the utility you get fom the outcome of schooling once you get the gaduation shock, given that the student s type is not evealed by a -shock. On the othe hand, it also inceases the option value of schooling and hence popensity to stay in school. These two e ects ae acting against eachothe..2 The Value-added of college We want to pice the option of going to college. Given maket incompleteness standad techniques ae not available. The appoach that we ae going to use follow fom the Equivalent Vaiation analysis famewok. Let be the maximum amount of wealth that a high-school gaduate is willing to pay to have the option of attending college. The condition fo can be witten as V x; w U W P (x ; p) 7 This esult is vey intuitive fo one of the potential extensions of the model whee we intepet to be a poxy fo cognitive abilities that di es among the students that is di eent, but not necessaily independent, fom being of type S o U. In paticula, we can assume that students with bette cognitive abilities lean faste about thei > 0 implies that students with bette cognitive abilities ae moe likely to enoll in and gaduate fom college.

12 also has the intepetation of the value-added of college education. Using the functional foms we obtain fo V (x; w U ) and W P (x; p), and solving fo, (p) f (p) w U Note that (p ) 0 because college has no value fo an agent with pio p..3 Time in school Anothe inteesting object is the expected amount of time in college fo any given individual with pio p 2 [0; ]. Fist note that an agent type S, i.e. p 0, the evolution equation fo his expected time in college is given by solving fo T (0), Fo an agent type S and p > 0, T (0) dt + dt 0 + ( dt) T (0) T (0) T (p) dt + dt 0 + ( dt) [dtt (0) + ( dt) T (p + p ( p) dt)] Taking limits when dt! 0, ( + ) T (p) + + p ( p) T 0 (p) Then, the expected time in college: 8 < p + p 0 p T (p 0 ) 0 p fo p 0 < p : 0 fo p 0 p The details ae in the appendix. Fo an agent type U the poblem is T U (p) dt + dt 0 + ( dt) ( dt) T U (p + p ( p) dt) 2

13 Afte taling limits, T U (p) + p ( p) T 0 U (p) The expected time in college fo an agent type U, 8 < p p 0 p T U (p 0 ) 0 p fo p 0 < p : 0 fo p 0 p 2 The poblem of a naive high-school gaduate In ode to evaluate the impotance of the option value of dop-out we solve a simila model as befoe but whee the dop-out option is not possible. Futhe, in this setup, agents does not account fo the leaning pocess. In some sense, we will be in pesence of myopic agents, whee the agent only takes into account the initial pio p 0 and then decides if it is woth it to attend college o not. The poblem hee, W no dop (x; p) max c e c + [pv (x; w U) + ( p) V (x; w S ) W (x)] (5) no dop +Wx (x; p) (x c a) Note that does not appea hee any moe because the chance of becoming a smat student is unimpotant fo this poblem. Poposition 4 (no dop-out) The solution to (5) is given by W no dop (x; p) e (x+g(p)) c x + g (p) whee g (p) is the solution to ( + (g (p) + a)) e g(p) pe Poof. see appendix. w U + + ( p) e w S + (6) Conjectue 2 Thee exists a theshold p such that an agent with pio p 0 p does not attend college. 3

14 Unde the conjectue, the VM condition fo this poblem is W no dop (x; p ) V (x; w U ). Lemma 4 g (p) > 0 and g 0 (p) < 0 fo p 2 [0; p ] Poof. Although the student does not have the option to dop out she can choose not to enoll in college ight afte high school. Theefoe W no dop (x; p) V (x; w U ) and hence g (p) > w U + > 0. Moeove, the left side of the equation (6) is deceasing in g (p) wheeas the ight side of the same equation in inceasing in p. Theefoe, g 0 (p) < 0. Using the VM condition we can solve fo p, In ode to have p 2 (0; ), p ( (w U + a)) e (w S w U ) e (w S w U ) ( (w U + a)) e (w S w U ) > 0 The value added of college in this case can be obtained fom the following equation: V x; w U W P;no dop x no dop ; p Solving fo no dop, Compaison of p and p. Fist we ewite p as no dop g (p) w U + p ( (w U + a)) e (w S w U ) + e (w S w U ) B e (w S w U ) + e (w S w U ) B (7) In ode to compae p and p st we need to sign e (w S w U ) B. Recall that h (B) ( + ln B (w S + a)) B 0. Also, because h is inceasing fo B > we have e (ws wu) > B i h e (ws wu) ( + (w s w u ) (w S + a)) e (ws wu) > 0 o, e (ws wu) e (ws wu) (w U + w a ) e (ws wu) > 0 (w U + w a ) > 0 4

15 Also, fom (3), [ (w u + a)] e (ws wu) > Now suppose e (ws wu) B:Then, we should have e (ws wu) (w u + a) 0 and B e (ws wu) 0 which contadicts (8) above. Then, B < e (w S w U ). De ne J e (w S w U ) B and ewite p as follows: p ( (w U + a)) e (w S w U ) + J e (w S w U ) + J ( (w U + a)) e (w S w U ) e (w + S w U ) + J! e (w S w U ) e (w p S w U ) + + J e(w S w U ) p + J e (w S w U ) + J B e (ws wu) (8) J e (w S w U ) + J J e (w S w U ) + J Given e (w S w U ) p < e (w S w U ) and J > 0, p > p As a esult we conclude that the enollment and gaduation ates should be lowe in the absense of the option to dop out of college. 3 Net Pesent Value Appoach The isk-neutal NPV appoach is widely used in studies of education choice. This appoach simply compaes income/cost ows. Fist we will conside the case that the agent can dop-out and then we will close this channel. 3. NPV with option of dop out De ne V NP V (w i ) ; W NP V;S and W NP V;P (p) as the net pesent value of being employed with constant wage w i, net pesent value of being a student 5

16 of type S, and net pesent value of being a student of unknown type. The NPV of an education can be constucted as follows: Staightfowad calculations povide: Also, Z V NP V (w i ) e t w i dt w i 0 W NP V;S w S a + ( + + ( p)) W NP V;P NP V;S (p) a + ( p) W + pv NP V (w U ) + ( p) V NP V (w(9) S ) NP V;P +Wp (p) p ( p) Conjectue 3 Thee exists a theshold p such that a student with pio p p dops-out of college. VM and SP conditions: W NP V;P (p) w U NP V;P Wp (p) 0 Poposition 5 (NPV value function) The solution to (9) is given by W NP V;P (p) w " + # S a + + ( p) p p p (w S w U ) ( p) + + p p + whee p [ (w S w U ) (w U + a)] [ (w S w U ) (w U + a)] + (w U + a) Poof. see appendix. In ode to have p 2 (0; ) it must be the case that (w U + a) < (w S w U ) (20) 6

17 3.2 NPV without option of dop out Staightfowad calculations povide, W NP V (p) a + p w U + ( p) w S ( + ) Conjectue 4 Thee exists a theshold p such that high-school gaduates with initial pio p 0 p do not attend college. The VM condition, Solving fo p, W NP V p V NP V (w U ) p (w S w U ) (w U + a) (w S w U ) (20) also guaantees p 2 (0; ). Compaison of p and p. Fist note that we can ewitte p as follows, (w S w U ) (w U + a) p (w S w U ) ++ (w U + a) So it is easy to see now that p > p, given that 4 Calibation of model: SATs ++ (w U + a) > 0. In ode to calibate the model we will match population moments. This means that we need to intoduce into ou pevious setup a notion of population, population types, and so on. We nomalize the total size of the high-school gaduates as a continuum ove the unit inteval. Also, let q U 2 (0; ) epesent the population mass of type U while q U ae of type S. We assume that the pio beliefs p 0 ae unifomly distibuted ove [0; ] fo each type. 8 In game theoy tems, 8 The identilca unifom distibution fo both types is a simply ng assumption. This can be substituted with anothe setting whee the distibution of type S agents is skewed to the ight elative to the distibution of type U in ode to captue a positive coelation of skill level and pio beliefs. 7

18 this means that in the st and second stage natue (N) plays and assigns to evey agent a type i 2 fu; Sg with pobabilities fq U ; q U g and then assigns a pio to each of them fom a unifom distibution between 0 and as illustated in the following gue. These two stages can be consideed as independent because of the identical unifom distibution fo both types of agents. As a esult, the existence of two types does not a ect the updating followed by the high-school gaduate so that the posteio belief conditional on q U is the same as the pio p 0. The assumption of same distibution fo p 0 fo both goups allows us to focus ou attention on the unobsevable e ects. If the distibutions wee di eent fom each othe, the agents could infe something about thei tue without any action and we want to put this possibility aside fo now. Afte Natue plays, the high-school gaduate, of unknown type i and known pio p 0 is pesented with the option of taking an exam that will povide a noisy signal about his tue type. Afte taking the exam, the high-school gaduate decides whethe to enoll in college o join the U- secto wokfoce. While in college he will lean about his tue type and will be allowed to dop-out if he decides so. The exam is cost fee and if the high-school gaduate ejects to take the exam he can t attend college. The gading of the exam is eithe pass (PASS) o fail (FAIL). Failing the exam also makes the high-school gaduate not suitable fo college education.the pobability of passing the exam if the tue type of the high-school gaduate is U is denoted by k U, while k S is the analogous fo the S type. In othe wods, P (P ASSjU) k U and P (P ASSjS) k S. Futhe, k S k U. Natue q u q u Natue Natue p 0 ~U[0,] p 0 ~U[0,] Type U Type S The next table is pat of Table of Navao (2005) 9. In 980, NLSY 9 page 74. 8

19 espondents wee examinated in Aithmetic Reasoning, Paagaph Composition, Wod Knowledge, Math Knowledge, and Coding Speed. All of these tests wee pat of the Amed Foces Vocational Aptitude Battey (ASVAB), and can be aggegated into the Amed Foces Quali cation Test (AFQT). As Navao (2005) points out, many papes have used this measue as a measue of scholastic aptitude. As it can be seen in the table, College gaduates scoed bette in evey subject than high-school gaduates that didn t continue thei education. In ou setup, this is captued by k S k U. We think of this exam as a poxy fo the SAT/ACTs. Povided the signal is not pefectly coelated with the tue type of the exam-take, the enollment ate can potentially incease and futhe the college completion ate can also decease. This is a esult of two di eent e ects: st, getting a F AIL in the exam automatically fobids the agent of attending college and second, povided getting a P ASS in the exam potential students update thei beliefs towads S. In this setup Bayes updating woks as follows: P (UjP ASS) P (P ASSjU) P (U) P (P ASS) k U p 0 P (P ASS) (2) o, using that P (P ASS) P (P ASSjU) P (U) + P (P ASSjS) P (S) we can ewite (2) as k U p 0 + k S ( p 0 ) P (UjP ASS) k U p 0 k U p 0 + k S ( p 0 ) 9

20 Note that P (UjP ASS) < p 0 so that a positive esult in the exam deceases the belief about likelihood of being type U. Calibation of Paametes. We set to be 0:03 so that e 0:97. Futhe, we set in ode to abstact fom intetempoal issues egading di eences between and. Using data fom the National Cente fo Education Studies (see Data Appendix), o NCES, we can obtain estimates fo the othe paametes. Befoe doing so, we have to think about what to look in the data. Agents with Associate degees and Vocational Ceti cates ae in esence, even though woking in the white-colla secto, unskilled agents. This implies that we have to adjust the data fo this fact (this is done in the Appendix). The estimates fo w U and w S and a (net cost of attending college) ae the following: w U w S 400 a 9934 To calibate we use estimates by Navao (2005). Navao estimates the coe cient of elative isk avesion,, using data fom NLSY. He estimates to be 2:5. The elative isk avesion with CARA utility function used hee is u 00 (c) c u 0 c (c) Then it follows that it should be the case o, c c Abstacting fom wealth 0, and povided, consumption by a student should be a convex combination between w U and w S. As a st ode appoximation, we assume that 2 (w S + w U ) 6:9 0 you can always think about wealth being close to 0 in ou model because of the inability of CARA models of geneating income e ects. 20

21 To calibate we look at the aveage expected time in college until completion. In data, an aveage student equies 4:58 yeas to leave college with a degee. Povided that the aival ate of eaning a degee is exponentially distibuted, the aveage expected time until completion is so 0:2834 Fo obtaining estimates fo, k U, k S and q U we have to equie fo numeical methods. Fo that, st we need to constuct a set of moments in ode to identify these paametes. Next we pesent these moments:. Popotion of agents woking in the unskilled secto: 0: Dop out ate: 0: College enollment ate: 0: Fom data we know that 49% of high-school gaduates takes the SATs (that ae equied fo admission into 4-y insititutions). Futhe, we know that 4-y college enollment is 0:435. The popotion of SATs takes that enoll in college is 0: Then, o, 4. Results P (P ASS) 0:88776 P (P ASSjU) q U + P (P ASSjS) q S 0:88776 k U q U + k S ( q U ) 0:88776 As a st step we set k S, that is, high-school gaduates of type S pass the exam with pobability. In the pevious section we poposed 4 moments and now hee we only needto estimate thee paametes. We will then use the exta moment to test the model. We will use moments 2 to 4 fo estimation and then moment fo testing. Using simulation we obtain that 0:263 k U 0:766 q U 0:4796 2

22 So ou simulated economy is populated fo a majoity of type S highschool gaduates. But, because is low, they will lean slowly about thei tue type and many will dop-out. In fact, we obtain that p 0:60289, so almost 40% don t attend college diectly. The college enollment geneated by the model is 58:8%, too high compaed with the 43:5% obseved in data appaently because the exam is not too infomative as is evident fom high value of k U. Dop-out ates ae in the simulation 24:6% vesus 32:5% in the data. Finally, the moment use as a test: in the simulation we obtain that the popotion of wokes in the unskilled secto is 7:5% a little highe than the taget value 66:7%. Clealy, thee is some oom fo impovement in ou calibation appoach. As an altenative method fo estimating the paametes (in contast of using simulation techniques) we can calculate the moments explicitely as functions of paametes and then solve them backwad in ode to get the paamete values. The details of this appoach which is still in pogess ae pesented in the Appendix 0.. Thesholds. Fo the fou di eent models: Also, p 0:60289 p 0:4025 p 0:7264 p 0:586 p p :4979 p p :2396 So NPV appoach oveestimates the tigge and ovestates the option value (it explains half of the tigge compaed to the case that accounts fo isk avesion). Futhe, imagine that thee is some population distibution F (p). Povided p > p, the isk-neutal NPV appoach will oveestimate the popotion of high-school gaduates that will enoll in college. Theefoe, iskavesion helps us emedy pat of the poblem associated with oveestimation of college enollment in the absence of psychic costs. Moeove, p p is almost twice as much as p p and p p is signi cantly bigge than p p so the NPV appoach undeestimates the e ect of the option value on college enollment ates. Befoe continuing with ou analysis of the value of college we need to appoximate f (p). Appoximation of f (p). 22

23 Recall that f 0 (p) (p ( p)) (p; f (p)). We can ewite this expession as f 0 (p; f (p)) (p) (22) (p ( p)) This equation will pove useful in a moment. Also keep in mind that f 0 (p ) and f (p ) ae known (by the VM and SP conditions). Say we ae inteested in the value f (p ). With this in mind, we do a Taylo expansion of f (p ) aound 0, f (p ) f (p ) + f 0 (p ) If we also want the value of f (p of f (p 2) aound 2, 2 2) we can do a Taylo expansion f (p 2) f (p ) + f 0 (p ) We can apply to this equation the fomula fo f 0 (p) that we have in (22), f (p 2) f (p ) + (p ; f (p )) ( (p ) ( (p ))) This suggests that we can poduce a ecusive algoithm to appoximate f (p) fo any p p in the following way: f (p (i + ) ) f (p i) + fo any i > 0. The next plot pesents the appoximation of f (p): (p i; f (p i)) ( (p i) ( (p i))) (23) emembe that it must be the case that p p. 2 we can also do the expansion aound 2 0 but in that case the quality of the appoximation will be educed. 23

24 Appoximation of f(p) 0.33 f(p) Value Matching p As discussed befoe it can be obeved that f (p) is positive and deceasing and that pastes smoothly into the outside option. 4.. Value of college The next plot pesents the value of college, as a function of the initial pio p 0, fo each of the di eent models. Fist, this gue shows that papes that assume isk neutality oveestimate the value of college because the cuves fo the isk neutal NPV agents lie above the cuves fo the isk avese agents. Moeove, college enollment and gaduation ates ae oveestimated in the isk neutal NPV case as is evident fom p > p. Second, the gue shows that the option value of leaning is much moe impotant when agents ae isk avese athe than isk neutal. These two esults imply that any seious model of college education should deviate fom isk neutality assumption and any model with isk avese students should take the option value of dopping out into account. 24

25 The value of college 2 CARA CARA naive NPV NPV naive hunded thousand U$S p 0 The lines ae fom top to bottom: NPV with option, NPV without the option, CARA with option and CARA without option. To quantify the main points of this pictue we nd: (a) college valoation in the uppe end (p 0 close to 0) is aound 200; 000 dollas in the fou cases, (b) NPV appoach ovestates the value of college fo as much as 00; 000 thousand dollas (this numbe is close to Cunha-Heckman (2007)), (c) the dop-out option seems to a ect much moe the theshold values of beliefs than the value of education. The next pictue allows us to undestand in a deepe way how the dopout option a ects the values. The pictue plots no dop and simila fo NP V appoach. 25

26 Size of the Option Value of dop out CARA NPV 0.25 hunded thousand U$S p 0 As it can be seen fom the gue the option value is much highe in the appoach that accounts fo isk (CARA). Fo example, fo p 0 0:5 the option value in isk avese case is moe than thee times the option value in the NPV case. Futhe, even though the option value does not explain that much of the total value (fom the pevious gue), the option value is moe impotant fo high values of p 0 and so it becomes of st ode impotance fo the theshold value of beliefs Time in college The black line is the expected time in college fo an S type agent with pio p 0 while the blue dotted line is fo a U type agent. The di eence between both lines accounts fo the fact that the S type has an exta channel explaining time in college: he can eceive a good signal and once this happens he will 26

27 not dop out. Expected time in college S U p 0 5 A Welfae Analysis and Policy With the infomation in hand we can un a countefactual analysis in ode to compae with Cunha and Heckman (2007) and Navao (2005). In paticula, we would like to compae the given famewok with a famewok whee all agents know thei type in ode to eliminate the welfae losses. We want to compute the ex-post eget of people that obtained a college degee and of the ones that didn t. Let R D be the popotion of agents that obtained a college degee and ex-post ealized that was bette fo them not to attend college. Accoding to ou model these people just nd out afte gaduation that they ae actually unskilled. Also, let R D be the popotion of agents that didn t get a college degee and eget that decision. These ae the skilled people that should have obtained a college degee but instead wok in unskilled secto. Using ou estimates we obtain that R D 26:8% R D 52% R D and R D ae a measue of welfae loss in this economy. Ou calculations suggests that thee is an impotant welfae loss. Futhe, these ough esults ae bigge to what Navao (2005) obtains (R D 0:2 and R D 0:25) and also fom Cunha and Heckman s (2007) esults (R D 0:38 and R D 0:75). The di eence fo the U-type agents is not that impotant povided that this agents ae only losing the opotunity cost. But, agents type S that not attend college o dop-out didn t obtain a high futue income steam and so thee is an impotant loss thee. 27

28 Even though still needs some wok, the methodology appeas to be pomising in the sense that povides a simple model that is able to match the moments obseved in data and futhe povide a simple tool fo public policy analysis. The policy implied by this analysis is to educe the ambiguity in students beliefs about themselves in ode to eliminate welfae losses. One way to implement this policy is to intoduce moe exams like SAT o ACT each of which will give infomation about the skill level of the students. 6 Extensions and Futue Wok Ou model povides a taceable famewok which can be extended to captue the popeties of the data bette by elaxing some assumptions. Fist of all, because the model assumes constant wages in both skilled and unskilled sectos and because the theshold value of beliefs do not depend on wealth, dopouts would not conside eapplying fo college at a late date. One way to allow fo etun to college is having stochastic wage pocesses in skilled and unskilled sectos so that students choose to go back to college if the gap between skilled and unskilled wages widens. This type of phenomenon has been obseved in the 80s. Second, the constant absolute isk avesion assumption leads to the simple popety of the model that the enollment and dopout decisions do not depend on the amount of wealth. Howeve, we obseve that the iche students ae moe likely to go to college and gaduate. We can use constant elative isk avesion utilitty function in ode to captue the di eence between ich and poo students. But this model is much less staightfowad that the CARA model. Fist, we do not have an explicit solution. Second, the e ect of wealth on the enollment and dopout decision cannot be qualitatively detemined at the beginning. In the CRRA case iche students will act as if they ae less isk avese in absolute tems wheeas the e ect of absolute isk avesion on the belief theshold can incease o decease with absolute isk avesion. Finally, because p does not depend on wealth in CARA case the students would not choose to e-ente the college late even if they had the option to do so. Howeve, in the CRRA case the student can choose to incease its wealth while not in college. This may in tun decease his theshold value fo > 0 and make him etun to the college. Thid, we can make the ate of aival of good news and gaduation shocks idiosyncatic and endogenous. Fo this pupose, we should sepaate woking skills and cognitive abilities. Students with bette cognitive skills 28

29 lean thei type faste and hence eceive the good news at a highe ate, conditional that they ae skilled wokes. We should also conside the possibility that the woking skills and cognitive abilities ae coelated. Moeove, we can think of the ate of gaduation shocks as inceasing with time o let it be di eent befoe and afte the eceipt of good news. This way, students that eceive good news stay less in college instead of lingeing aound campus fo a long time. We can endogenize the aival ate of these shocks by letting the students adjust them at some monetay o utility expense. Fouth, we can povide some empiical content fo the initial pio p 0. Imagine a model whee p i 0 (X i ) + " i whee X i is a vecto of obsevables chaacteizing high-school gaduate i and " i is the unobsevable pat (both to the econometician and the highschool gaduate) such that X i? " i, E (" i ) 0 so " i is unfocastable. X i can include the GPA aveage, income of paents, education of paents, and so on. We can un a egession on obseved wages (that we thing is a good pedicto of ability) on this chaacteistics in ode to compute ^p i 0. Then, we can ty to gue out the distibution of " i that bette ts the data. Finally, we can exploit the dynamic natue of ou model to t the tajectoies of di eent vaiables ove time as a obustness check. 7 Conclusion In this pape, we focus on college enollment and dopout decisions in a taceable options model whee the psychic cost stems fom the initial belief of students about thei skill level, poviding a less contovesial explanation fo the once obscue psychic costs. In this model the two main bene ts of college education ae cedential e ects and leaning. Ou pape has thee main esults. Fist, ou model shows that papes that assume isk neutality oveestimate the value of college, college enollment and gaduation ates because the outcome of college education is isky. Second, we show that the option value of leaning is much moe impotant when agents ae isk avese athe than isk neutal. Futhe, the e ect of the option value is much moe impotant at the magin so it becomes impotant fo undestanding dop-out ates. These esults imply that any seious model of college education should deviate fom isk neutality assumption and any model with isk avese students should take the option value of dopping out into account. 29

30 We also show that standaized tests, noisy measues of own ability, can be undestood as a welfae impoving policy as they can educe uncetainty about futue income steams. Futhe, such a policy deceases college enollment and inceases gaduation ates as moe bad apples will be eliminated befoe enolling in college. Finally, we poduce a counte-factual analysis wee we eliminate all uncetainty in the model. We nd that thee is an impotant welfae loss in the economy due to uncetainty. Futhe, we show that uncetainty hits skilled agents hade. 8 Refeences. Altonji, Joseph G. (993). "The Demand fo and Retun to Education when Education Outcomes ae Uncetain", Jounal of Labo Economics, Vol., No. 2. Belley, Phillipe and Lochne, Lance (2008). "The Changing Role of Family Income and Ability in Detemining Educational Achievement", Jounal of Human Capital, Vol., No.. 3. Caneio, Pedo, Hansen, Kasten T. and Heckman, James J. (2003). "Estimating Distibutions of Teatment E ects with an Application to the Retuns to Schooling and Measuement of the E ects of Uncetainty on College Choice". Intenational Economic Review, Vol. 44, No Comay Y., Melnik, A. and Pollatschek, M. A. (973). "The Option Value of Education and the Optimal Path fo Investment in Human Capital", Intenational Economic Review, Vol. 4, No Cunha, Flavio and Heckman, James J. (2007). "Indetifying and Estimating the Distibution of Ex Post and Ex Ante Retuns to Schooling", Labou Economics, Vol. 4, No Dixit, Avinash (993). "The At of Smooth Pasting", Fundamentals of Pue and Applied Economics, No Hansen, Las P. and Sagent, Thomas J. (200). "Robust Contol and Model Uncetainty", AER. 8. Heckman, James J. and Navao, Salvado (2006). "Dynamic Discete Choice and Dynamic Teatment E ects", Jounal of Econometics. 30

31 9. Heckman, James J., Stixud, Joa and Uzua, Segio (2006). "The Effects of Cognitive and Noncognitive Abilities on Labo Makets Outcomes and Social Behavio", Jounal of Labo Economics, Vol. 24, No Judd, Kenneth L. (2000). "Is Education as Good as Gold? A Potfolio Analysis of Human Capital Investment", unpublished, Hoove Institution.. Keane, Michael P. and Wolping, Kenneth I. (997). "The Caee Decisions of Young Men" Jounal of Political Economy, Vol. 05, No Miao, Jianjun and Wang, Neng (2007). "Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival", unpublished. 3. Navao, Salvado (2005). "Undestanding Schooling: Using Obseved Choices to Infe Agent s Infomation in a Dynamic Model of Schooling Choice when Consumption Allocation is Subject to Boowing Constaints". unpublished, Univesity of Chicago. 4. Stange, Kevin (2007). "An Empiical Examination of the Option Value of College Enollment". mimeo, Univesity of Califonia Bekeley 9 Appendix 9. Poof of Poposition (Value at Wok) V (x; w i ) max c The FOC wt c implies Plugging into (24), e c V (x; w i ) V x (x; w i ) e c c + V x (x; w i ) (x + w i c) (24) V x (x; w i ) ln V x (x; w i ) + V x (x; w i ) x + w i + ln V x (x; w i ) 3

32 Guess V (x; w i ) e (x+w i+a) Solving fo A, Thus, e (x+w i +A) and evaluate the last equation, e (x+w i+a) +e (x+wi+a) (x + w i (x + w i + A)) A e x+w i V (x; w i ) Using the FOC, c (x; w i ) x + w i > is needed follows fom the fact that othewise the tansvesality condition will not hold (the agent can save evey peiod his whole income and consume at t ). The tansvesality condition is lim e t jv (x; w i ) j lim e t j e t! t! x+w i j Poof of poposition (Value of an S type student) (2) can be witten as ( + ) W S (x) max u (c t ) + V S (x; w S ) + W c x S (x) (x c t a) t e ( + ) W S c (x) max c e x+w S + Wx S (x) (x c a) The FOC wt c implies, Substituting back, ( + ) W S (x) W S x e c c e x+w S 32 Wx S (x) ln W x S (x) +Wx S (x) x + ln W x S (x) (25) a

33 Guess that the value function is of the fom W S (x) B e (x+w S+A) Now, apply the guess into (25), ( + ) Be (x+w S+A) B e (x+w S+A) +Be (x+w S+A) e x+w S x + ln Be (x+w S+A) a guess that A. Then the equation can be educed to, it can also be witten as, ( + ln B (w S + a)) B ln B B + (w S + a) Finally, c ln W x S (x) ln B + x + w S x a B Povided that B exists we can check that the tansvesality condition holds: lim e t jw S (x) j lim e t j B e (x+ws+a) j 0 t! t! 9.3 Existence and Uniqueness of B The next two lemma summaizes the popeties of B. Lemma 5 (Existence of B) Thee exists B > such that (4) is satis ed. Poof. De ne h ~B + ln B ~ (w S + a) ~B is continuous. Futhe, h () (w S + a) < 0. Note that h ~B 33

34 And, lim h ~B + ~B!+ Then, by the mean-value theoem thee exists a B 2 (; +) such that h (B) 0. Lemma 6 (Uniqueness of B) B is unique. Poof. Note that lim h ~B ~B!0+ also, h 0 B ~ (w S + a) + + ln B ~ and, lim h 0 B ~ < 0 ~B!0+ We also have that h 00 B ~ > 0 B ~ This implies that h ~B is stictly convex. It is deceasing fo B ~ < B whee h 0 B 0 and inceasing fo B ~ > B. Moeove, h B < 0. Theefoe, thee exists a unique B such that h (B) 0. The next gue summaizes the popeties of h ~B, h ( B ~ ) φ B B ~ 34

35 Coollay The fact that h (B) 0 implies that Then, ( + ln B (w S + a)) B > 0 ln B > (w S + a) B > e (w S+a) and thus, B 2 max ; e ((w S+a) ) ; + This coollay will show to be useful fo di eent calculations. 9.4 Compaative statics fo the value of an S type student Fist we need to undestand how B changes with the di eent paametes. Then, Note that, db B dw S + + ln B (w S + a) db B da + + ln B (w S + a) db (w S + a) B ln B d + + ln B (w S + a) db d B + + ln B (w S + a) db (w S + a) B d + + ln B (w S + a) ++ ln B (w S + a) B + ( + ln B (w S + a)) B B B + B > 0 35

36 and so, db dw S B2 B + > 0 db da db d db d db d db d B2 B + > 0 (w S + a) B ln B B+ B ( B) B B + < 0 (w S + a) B 2 B + 0 > 0 ( B) (w S + a) B + B + > Poof of poposition (Value of an unknown student) (5) can be witten as ( + + ( p)) W P (x; p) e c max + ( c p) W S (x) + [pv (x; w U ) + ( p) V (x; w S )] The FOC wt to c povides +W P x (x c a) + W P p (p ( p)) e c c W S x ln W x S Plugging into the oiginal poblem, ( + + ( p)) W P (x; p) W S x ( 2 4p e p) B e x+w U x+w S + ( p) e x+w S 3 5 +W P x (x c a) + W P p (p ( p)) 36

37 Guess that W S (x; p) Afte some algeba, ( + + ( p)) e f(p) e (x+f(p)) e f(p) 2 + 4p e and apply into this last equation. + ( p) B e w U w S + ( p) e w S 3 5 o +e f(p) (f (p) + a) e f(p) f 0 (p) (p ( p)) + + ( p) + f 0 (p) (p ( p)) (f (p) + a) e f(p) ( p) Be w S + pe w U + ( p) e w S 9.6 Compaative statics fo p e (w S w U ) + db da ( + ) e (w S w U ) B db ( + (w U + a)) e (w S w U ) B da ( + ) e (w S w U ) B 2 e (w S w U ) + db da ( + ) e (w S w U ) B db ( + (w U + a)) e (w S w U ) B da ( + ) e (w S w U ) B 2 we know that ( + (w U + a)) e (w S w U ) B > 0 and so ( + ) e (w S w U ) B > 0. Also we know that db da > < 0 37

38 e (w S w U ) B 0 (+)e (w S w U ) B (+ (w U +a))e (w S w U ) B h i (+)e (w S w U ) 2 B e (w S w U ) B ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) C A e (w S w U ) B ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) We have that h (B) ( + ln B (w S + a)) B 0. Also, because h is inceasing fo B > we have e (ws wu) > B i ( + (w s w u ) (w S + a)) e (ws wu) > 0 e (ws wu) (w U + w a ) e (ws wu) > 0 e (ws wu) (w U + w a ) > 0 Also, fom (3), [ (w u + a)] e (ws wu) > Now suppose e (ws wu) B:Then, we should have e (ws wu) (w u + a) 0 and B e (ws wu) 0 which contadicts (26) above. Then, it > 0 B e (ws wu) (26) e (w S w U ) db d 0 B (+)e (w S w U ) (+ (w U +a))e (w S w U ) B h i (+)e (w S w U ) 2 B e (w S w U ) db d ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) > 0 C A 38

39 9.6.4 Wage in the S (w U + a) e (w S w U ) ( + ) e (w S w U ) B ( + ) e (w S w U ) db dw S + ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) B ( + ) e (w S w U ) db dw S ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) db + B + dw S B B +! ( + ) e (w S w U ) B (w U + a) e (w S w U ) ( + ) e (w S w U ) B Wage in the unskilled (w U + a) e (ws wu ) e (ws wu U ( + ) e (w S w U ) B (w U + a) e (w S w U ) ( + ) e (w S w U ) B 2 > 0 ( + ) e (w S w U ) ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) " # (w U + a) ( + ) (w U + a) e (w S w U ) ( + ) e (w S w U ) B ( + ) e (w S w U ) B 2 e (w S w U ) " # (w U + a) (B + ) ( + ) e (w S w U ) B 2 + ( + ) e (w e (w S w U ) S w U ) U < ( + ) e (w S w U ) B + db d ( + ) e (w S w U ) B 2 (w U + a) e (w S w U ) 39