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FINANCIAL AID TO GRADUATE STUDENTS: AN HYPOTHESIS FOR APPLIED PRICE DISCRIMINATION

R. A l len Moran

An issue which is now drawing cons iderab le a t t e n t i o n on the par t o f co l lege a d m i n i s t r a t o r s is the mat te r of whether s c h o l a r - ships to undergraduate_ s tudents ought to be awarded on the basis of s tudent a b i l i t y r a the r than (or in a d d i t i o n to) f i n a n c i a l need. While t h i s is a r e l a t i v e l y new issue fo r undergraduate educa t i on , i t is a mat ter which has long posed a dilemma fo r those whose task i t has been to award f e l l o w s h i p aid at the graduate l e v e l . Data on the sub jec t being h igh l y f u g i t i v e , i t is t h i s w r i t e r ' s i n t u i t i v e f e e l i n g t ha t most awards at the graduate leve l have, u n t i l r e l a t i v e l y r e c e n t l y , been made l a r g e l y on the basis of s tudent a b i l i t y .

Whatever the p r e v a i l i n g p r a c t i c e , however, the issue t ha t is germane is the quest ion o f how such awards ought to be made. Presumably i t is poss ib le to de f ine an optimum usage of graduate educat ion f a c i l i t i e s and to t a i l o r the g ran t i ng o f f i n a n c i a l aid toward ach iev - ing t h i s optimum. Economists have given some a t t e n t i o n to t h i s quest ion from a s o c i e t a l v i ewpo in t , but r e l a t i v e l y l i t t l e e f f o r t has gone in to the ana l ys i s of the e s s e n t i a l l y managerial opera t ions of the i n d i v i d u a l academic department . Unless the ope ra t i ona l goals o f the department can be s p e c i f i e d , one con f ron ts a problem akin to t h a t of d iscuss ing the a c t i v i t i e s of f i rms w i t hou t a wel l developed theory of the f i r m . To r a t i o n a l l y descr ibe the a l l o c a t i o n of f i n a n c i a l aid requ i res the development of a c r i t e r i o n f u n c t i o n fo r the academic depar tment . 1

The model proposed here is something o f a d e v i l ' s advocate 2 c o n s t r u c t i o n in t ha t the goal pos tu la ted f o r the academic department represents a view of depar tmenta l p re ferences t ha t few a d m i n i s t r a t o r s would be w i l l i n g to defend in p u b l i c . Simply pu t , l e t us suppose tha t the p r o t o t y p i c a l academic department seeks to maximize i t s " p r e s t i g e " and t ha t t h i s p res t i ge is to at l e a s t some degree re l a ted to the c a l i b e r of i t s graduate s tudents as r e f l e c t e d in the3Job placement they obta in a f t e r t h e i r tenure in the department . The o b j e c t i v e o f f o r m u l a t i n g the model is a modest one and designed to s t i m u l a t e f u r t h e r d iscourse on the sub jec t of f e l l o w s h i p awards r a t h e r than to prov ide t i d y answers to d i f f i c u l t ques t ions . What we seek here are the i m p l i c a t i o n s of the " p r e s t i g e " hypothes is w i th respect to the g ran t i ng o f f e l l o w s h i p s on the basis of f a c t o r s o ther than i n t e l l e c t u a l m e r i t . The value of the exerc i se l i e s in the develop- ment of i n s i g h t i n to the t h e o r e t i c a l aspects of a problem of manager ia l choice which academic economists are o f ten ca l l ed upon to make.

The approach to be used is one in which the graduate program uses s tudents (o f d i f f e r e n t a b i l i t i e s ) as inputs (w i th d i f f e r e n t p r i c e s ) in the p roduc t ion of f i n a l outputs which convey bene f i t s to the educat ing i n s t i t u t i o n ( the academic depar tment ) . The outputs thus take the form of completed Ph.D. 's placed in var ious j obs , l eve l s o f complet ion shor t o f the te rmina l degree, e tc . To each of these outputs is a t tached a b e n e f i t measure which may vary w i th the p a r t i c u l a r graduate i n s t i t u t i o n . The problem, then, is to maximize the b e n e f i t to the educat ing i n s t i t u t i o n sub jec t to c e r t a i n c o n s t r a i n t s

The p r o t o t y p i c a l educat iona l " f i r m " is conceived of as a graduate department in a u n i v e r s i t y which operates both a graduate and an undergraduate program. The framework in which the ana lys i s takes place is s h o r t - r u n , so a number of s i m p l i f y i n g assumptions

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are possible:

(1) Standards for the completion of the Ph.D. (and other) degree(s) are assumed constant.

(2) Faculty size, the dimensions of the undergraduate program, and research commitments of faculty time are assumed fixed.

(3) The amount of fellowship funds to "buy" graduate students is treated as a constant.

(4) The university competes for i ts graduate students with other "employers" whose actions are independent of i ts own.

(5) The incremental benefit to the department of placing a student in a given situation is assumed constant.

(6) The probabil i ty that a student with given ab i l i t i es can achieve a given level of completion (and then earn a part icular placement) is estimable. 5

The problem in terms of a department's graduate admissions policy may then be specified as follows. The entering class is chosen which maximizes the expected total benefit ("prest ige"), U, subject to a fixed amount of fellowship funds, F o. U is then given by

n U = Z xiu i i = 1,2, . . . , n

i=i for n categories of students (by ab i l i t y - - the ab i l i t y of the i th category is assumed to be superior to that of th~ i + I s t ) , where ~l is the number of students admitted from the �9 h category and u i is the expected benefit a student from the i th category yields to the ins t i tu t ion , v

Since our typical department must compete with other departments for i ts graduate students, i t may be assumed that the number i t gets from a given ab i l i t y category wi l l be dependent (in the short-run framework given) on the fellowships i t awards (and tu i t ion charged, which shall be assumed fixed) so we may write 7

I

xi= gi ( f i ) i = 1,2, . . . , j , k , . . . ,n; gi~O

where f. is the fellowship to be given to the i th category of student. T~e department then seeks to maximize expected benefit subject

to the constraint imposed by limited fellowship funds. The Lagrangian expression for the department's act iv i t ies may be expressed by

V = U - ~(~ x I f i - FO) i=l

to be maximized by the choice of n d i f ferent levels of fellowship aid. Assuming second order conditions are met, the f i r s t order conditions for a benefit maximum are

r x i) =0 for x i>O ~V = g~ ui -~x(f i + gl .p for x i =0 i= 1 , 2 , . . . , j , k , . . . , n ~ f i n

and ~- x i f i = Fo" i=l

To i l l us t ra te the implications of the model for the "need versus merit" issue, consider two classes of students, the j th and the k th, where, by assumption, the former have more ab i l i t y than the la t te r . The f i r s t order conditions stated above may be rearranged to show that at a prestige maximum

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f j uj ( l § k

fk u k (I + I / e j ) I

where e = ( f ~x.) ~gi ) f o r i 1 , 2 , . . . , j , k , . . . , n is the e l a s t i c i t y of suppiy of ~u~en~s in the i ~h category wi th respect to f e l l o w s h i p money provided to them and in turn r e f l e c t s the e l a s t i c i t y of demand of t ha t class of s tudent fo r our depar tment 's educat iona l se rv i ces .

I f the o b j e c t i v e is to maximize departmental p r e s t i g e , and i f i t is opt imal to give fe l l owsh ips on the basis of mer i t a lone, then i t should f o l l ow in the contex~ of the present example tha t f j / f k ~ I . But t h i s is not the case. he assumptions made requ i re tha t u i / u k > l , but f J / f k is also a f f ec ted by the e l a s t i c i t i e s of s tudent supp ly , ej and e k. Put d i r e c t l y , i f the supply of (some category o f ) poorer students is h igh l y e l a s t i c (or i f the supply of some ca tegor ies of b r i g h t e r students is h i gh l y i n e l a s t i c , then the poorer students ( in terms of a b i l i t y ) may receive the l a rge r f i n a n c i a l awards. This represents a l o g i c a l extension of the concept of monopoly p r i c i n g t ha t buyers wi th a more i n e l a s t i c demand pay, ce te r i s pa r ibus , h igher p r i ces .

I t is c l e a r l y unwise to put too much weight on a t h e o r e t i c a l model unbacked by empi r i ca l evidence, but the conc lus ion reached above suggests some though t fu l specu la t ion may be in order . Suppose, fo r example, i t were to be found tha t marr ied women were o f f e red smal ler f e l l owsh ips than men. Such a s ta te of a f f a i r s need not be based on any not ion tha t women are of i n f e r i o r mental f i b e r nor even on the idea tha t they are less l i k e l y to complete a degree and thus less l i k e l y to c o n t r i b u t e favo rab ly to the p res t i ge of the department. Marr ied women might s imply be given smal ler f e l l owsh ips on the pre- sumption tha t they are supported by husbands who hold jobs they regard as s a t i s f a c t o r y . Since the wives then become to an ex ten t g e o g r a p h i c a l l j immobi le, t h e i r demand fo r the serv ices of a degree program (and thus the supply of t h e i r persons as s tudents) tends to be h i gh l y i n e l a s t i c . By s i m i l a r l o g i c , e x c e l l e n t students from small ru ra l co l leges m~ght be given lower f e l l o w s h i p o f f e r s from t h e i r s ta te u n i v e r s i t i e s ~ a n less accomplished students from Ivy League schools , not because the fo rmer 's c r e d e n t i a l s are suspect, but because the l a t t e r are seen as more mobile and thus in more e l a s t i c supply. To the ex ten t t ha t f i n a n c i a l need a f f e c t s the e l a s t i c i t y of supply of d i f f e r e n t ca tegor ies of s tudents , i t becomes a c r i t e r i o n fo r awarding f i n a n c i a l a id .

The o b j e c t i v e of t h i s commentary has not been to e i t h e r condone or condemn the p res t i ge hypothes is , but ra the r to suggest some of the ways in which the adopt ion of such a p o l i c y by a department would a f f e c t the a l l o c a t i o n of f e l l o w s h i p funds among graduate s tudents . Since academic economists seem to be almost un i fo rm ly concerned wi th depar t - mental " p r e s t i g e " , and since i n f l a t i o n is making educat iona l i n s t i t u t i o n s i n c r e a s i n g l y conscious of f i n a n c i a l c o n s t r a i n t s , the time seems r i g h t fo r a d ia logue on the i n t e r a c t i o n of the depar tment 's managerial goals and the choice of students to receive f i n a n c i a l ass is tance.

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I . In a very real sense the problem here is one of sub-optimization since the problems of managerial choice involve only the graduate program and even then only choices made at the department level.

2. The following analysis wi l l be couched in terms of a Ph.D. program in economics.

3. Such an assumption buries some very nice issues. For instance, i f prestige is unrelated to student qua l i ty , the result ing model with respect to the choice of graduate students is an "a-body-is- a-body" model in which the department takes in large numbers of students and lets them fend for themselves. I f "prestige" is unrelated to job placement and can be measured simply by the proportion of students completing a program i t is easy to see how the department could accomplish this goal un i l a te ra l l y .

4, The model could easily be altered to have graduate students serve as teaching assistants.

5. This assumption is heroic, but i t allows one to concentrate on the managerial problems at hand. Presumably estimates of this sort must be made in any event.

6. Each u i in turn is derived as follows:

m m

ui j~-Zl PijWj ( j= l ~ Pij l )

where m is the number of possible states of completion (ranging from semester drop-out through, e._~_~_, placement teaching t Harvard); ~ is the probabi l i ty that a student in the .t~ category wi l l gain ~ j th placement, as a result of graduate study; and w- is the benefit of the j~h placement to the department (in terms o~ "prestige" increment) where a par t icu lar wj may be posi t ive, negative, or zero.

7 i f tuition becomos a variable, one could w ite x.; i where t is t u i t i on , etc. I t is here assumed tha~ a l . f : . ,owships are posit ive or zero.