Teaching Assessment Trade-off and Performance Related Pay in Education
Rajlakshmi Mallik
and
NSHM Business School, Kolkata, India
Diganta Mukherjee*
ICFAI Business School, Kolkata, India
Abstract
Teacher's allocation of effort between teaching and assessment and its consequences on human capital
formation has caught the attention of academic researchers for some time. This paper develops a
model in a principal agent framework to study the interaction between the governing body, the teacher
and the student of an academic institution. The choices of unobservable effort levels by the teacher
and the student for the relevant activities can be manipulated by the governing body by choosing
appropriate incentive mechanisms. In a sequential move game, we solve for all such mechanisms by
the method of backward induction and establish necessary and sufficient conditions for these
mechanisms to be equilibrium behaviour under alternative payoff structures. The trade-off between
teaching and assessment effort is sharply delineated in this model.
Key Words: Teaching, Assessment, Performance related pay, Principal agent model
JEL classification numbers: D82, I21, I22
*
Corresponding Author: Diganta Mukherjee, ICFAI Business School Kolkata, Plot J-3, Block GP,
Sector V, Salt Lake City, Kolkata - 700 091, India, [email protected], [email protected]
1. Introduction
The link between education and employment is crucial for economic development. Highly skilled
graduate labour constitutes one of the most important factors of production for enterprises that are
pivotal to economic growth. Education generates this human capital by enhancing an individual's
embodied skills above their raw labour ability (Belfield and Levin, 2003). Empirical research has
found evidence of a strong positive relationship between earnings and higher education indicating the
quality enhancing role of the educational process (Cohn and Addison, 1998). It is this economic
benefit of higher education that explains why students enrol with institutions of higher education and
government funding of higher education is substantial (Goldin and Katz, 1998).
Given its importance, question arises as to what are the characteristics of the production that takes
place in institutions of higher education. Rothschild and White (1995) and Dolan et. al. (1985)
stress on the joint role played by students (willing to invest their time and effort to learn skills) and
teachers (who have the skill and are able to impart the training) in the generation of human capital.
Among other inputs, one that is less frequently mentioned is the role of university managers who
influence the technical efficiency of the universities by allocating resources within the institution and
acting as principals in structuring incentives for the agents (Johnes, 1999). Among the various
educational policy reforms one that has attracted a lot of interest pertains to the policy of performance
related pay (PRP) for teachers both at school level and in higher education. A large number of
empirical studies find evidence for a positive impact of teacher salary levels on students’ outcomes at
the school level (Loeb and Page, 1999, Dewey et al, 2000). Evidence also reveals that teacher pay
affects teacher performance by influencing the recruitment and retention of more able teachers
(Jacobson, 1995, Dolton and Van der Klaauw, 1996, 99) but more importantly by inducing better
performance by continuing teachers (Hanuhek et al, 1999). Using this evidence and from the
preceding discussion on the characteristics of higher education we find that the empirical basis for
developing a principal-agent model of education for analysing the impact of PRP for teachers in
higher education is significant.
We consider the Governing Body (GB) of an academic institution as the principal. There are two
types of agents: teachers and students. The GB wants the agents to perform two activities resulting in
two observable outputs - realised output as an indicator of skill formation and certificates of student
effort. The first activity, skill formation, involves effort by both the teacher and the student who must
incur effort in teaching and learning respectively. The agents' efforts are not observable to the GB.
Thus we have a principal-agent set up with joint production by multiple agents where agents' efforts
can not be distinguished. The other activity, namely assessment leading to certification, requires effort
by only one of the agents, the teacher. This introduces an element of multi-tasking into our principalagent framework with multiple agents. We assume that all efforts are costly to the agents. The GB
benefits if high quality output is realised and certificates match realised output. We assume that the
GB does not directly pay a wage to the student. The student receives his pay from some outside
source, say, the firm according to some exogenously given wage schedule and the certificate of effort
issued by the teacher. The teacher receives his pay from the GB. The GB's problem is then to design a
pay schedule for teachers such that it induces high effort by teacher in both activities - skill formation
and assessment. The former would increase probability of high output being realised, which is
beneficial for GB. The latter will induce high effort by student in learning and also increase
probability of match of certificates with realised output and thereby benefit the GB on both counts.
Teacher assessment will promote student effort in this environment.
Thus our model is able to capture certain aspects of teaching, namely multi-tasking and joint
production, with which PRP is deemed to be non-amenable in the literature (Holmstrom and
Milgrom, 1992, Bernheim and Whinston, 1998, Hannaway, 1992). This paper does not address the
issue of substitutability or complementarity between efforts linked to different tasks. The two tasks
are independent in terms of effort cost. It does however make structural assumptions regarding the
relative disutility from effort in teaching and assessment. Typically, teachers are less fond of
assessment than they are of teaching. Under the given structural assumptions we try to locate the kind
1
of equilibria that might prevail under different award schemes. Specifically we address the issue when
are we likely to observe equilibria with either low teaching or low assessment effort rather than
equilibria with high effort in both. Further, when are we likely to observe low effort in teaching rather
than in assessment and vice-versa. This would enable us to draw some policy inferences regarding the
type of incentive structure that might promote socially desirable equilibria.
The preferences and costs of the agents, i.e. the teacher and the student, are described in the next
section. The principal - agent interaction is modelled as a sequential move game where the principal
moves first by announcing the payment scheme and the agents subsequently choose their effort levels
for relevant activities. This game is described along with the payoffs. We proceed by backward
induction. Given the GB's choice of pay schedule, we identify the equilibrium strategies for the agents
and demonstrate the necessary and sufficient conditions for the existence of such equilibria. Section 3
then looks at some of the comparative static exercises in numerical terms, as these are non-amenable
to algebraic treatment. Concluding remarks are made in section 4.
2. The Model and Results
Describing the Model
We now make a formal statement of the model described in the introduction. We consider a set up
with three participants, the governing body (GB) of an institution of higher learning 1, a teacher and a
student. The GB and the teacher are directly tied in a principal - agent relationship. The GB recruits
the teacher to perform some activities which involves effort on part of the teacher. The GB's payoff
also depends on the action of the student which involves effort as well. The GB however does not
directly interact with the student. The pay that the student receives as an employee of a firm serves as
the incentive2. Thus in our model, the student's choice of action is influenced by the wage schedule
offered by the firm and the teacher's actions. The GB does not monitor the student directly but only
indirectly. Moreover, we assume that the wage schedule faced by the student is exogenously given3.
Thus although the student receives pay from a firm, the firm plays a passive role in this model. In
other words this model does not consider any principal - agent relationship between the firm and the
student. The following subsections describe the activities, cost and pay functions for the teacher and
the student.
The Teacher chooses teaching effort, t, simultaneously with student choosing the learning effort, e.
Only the teacher can observe t and e separately. The resulting output, student quality is observable to
all and is given by the following relation:
q  Q(t , e)  te .
The teacher subsequently puts in the assessment effort, a, and awards the student grade, g, according
to the following relation:
g  G ( q, a ) ,
which is public information. Finally, the placement happens with the package the student receives
being determined from q according to the relation:
1
The academic institution we consider is an institution of higher education (e.g. for post - graduate study, like
universities, research institutes or institutions imparting special professional training like Medical, Law,
Engineering etc.) where the student input is very important, as discussed earlier and formalised below.
2
We have deliberately kept firm's training out of the picture to delineate the teacher's effect on productivity.
This is a departure from Spence (1974). In this model we assume that the probability of finding a job is one for
those with training and zero for others.
3
We can think of the prevailing labour market wage schedule as the exogenously given wage schedule.
2
   ( q,  )   
0
1 1
1 ,
( q )  
te
 2
2
2
0
where  is the (im)precision parameter of the market valuation of student quality.
The teacher’s (dis-)utility from the teaching and assessment effort is given by:
1
1
T (t , a)   a  t , 0    1.
2
2
2
2
So we are assuming that assessment requires a higher rate of effort than teaching; in other words, the
teacher likes to teach more than assessing.
The increment in the teacher’s incentive package (S) depends directly on two factors, placement  and
discrepancy between placement and the grade the teacher awarded to the student, D, which is
governed by the relationship:
D

d (q) 
a

a
 g 

a
q
0
 q  q .
0
Here, the discrepancy is benchmarked by a tent shaped function of q, which increases from 0 linearly
for
q  q , reaches a maximum of
0
q
0
and then decreases linearly to 0 again. This is representative
of the idea that it is more difficult to assess mediocre ability than very good or very bad cases. We are
assuming that q is a suitable middle value. Also, it is natural to assume that the discrepancy is
0
mitigated by a higher level of assessment effort, a, and increases when the placement package
determination is more imprecise ( is larger).
The final form for the increment is:
S  S    D ,
t 1
where
t
S
t
is the teacher’s pay at time t. The importance of placement is determined by the parameter 
t
and that of discrepancy by . Taking into consideration the teacher’s disutility from teaching and
assessment, the net payoff for the teacher is:

=
S  T (t , a)
t 1
=
S    D  T (t , a)
t
=
t
1 

1
1

S   
te    q  q  q   a  t
2 
a
2
2

2
t
0
0
2
0
The student’s net payoff, from placement package and disutility from learning effort, is given by:
    c ( e) 
t
1
1
te  e
2
2
2
.
The Effort Choice Problem
We now proceed to solve the effort choice problems of the teacher and student. We will proceed by
backward induction, starting with the second stage where given (t, e), the teacher chooses a. The
optimal choice is given by the solution of the following first order condition:



d (q )  a  0 , which implies the optimal value
a
a
a* 
3
d (q ).
2
3
The second order condition is satisfied as:

2

d (q)  1  0 .
a
a
2
2
3
Now we will solve the first stage problem of teacher and student simultaneously choosing t and e. The
corresponding equations are:
t ………………………….. (1)

1

t  e  0 , implying e* 
2
e 2
Also  
, thus the solution in (1) is indeed a maxima.
2
e
2
 1  0
Thus we see that learning effort is always directly proportional to teaching effort in our set up.
Also, in equilibrium
t*
2
q*  t * e * 
………………………….(2)
So again, quality is always directly proportional to teaching (or learning) effort in our set up.
And for the teacher,
 (a*)

3
1 e

e  ( ) (d (q )) I (q  q )
 t  0 …………(3)
t
2
2
2 t
2
1
3
3
0
Combining the reaction functions of the student and the teacher, equations (1) and (2), we obtain the
following equilibrium condition:

3
1 1
t  ( ) ( d ( q )) I ( q  q )
 t  0
4
2
2 2
or t  3   d ( q ) I ( q  q )  t  0 ………………………….(4)
4
4 2
2
1
3
3
0
2
2
3
1
1
6
3
0
2
Now, depending on the value of q, we have to consider two cases separately.
Case 1:
qq .
0
This is equivalent to saying that
t
q
2
tq
or
0
t
3

 
4
4 2
3

Which implies  
  t 
   2q

 4
 4 2

Then the equation (3) becomes
2
3
1
1
3
1
0
For a solution to the above equation, we need
3
6
1
6
2
t 

,
 2q 
  t  0
2 

t 
.

 0
2 
0
2
2
2 .
0


4
3
(otherwise, both terms on the LHS will be
2
positive). Then the solution is given implicitly by
t* 

t *  2q 

2 

0
1
3
3 
2


3
21
..................(5a)
6
4 2   
2
4

4
e* 
and then
a* 
3
 (2q  q)
t*,
2
(again implicitly defined though q*). So in the high quality situation,
0
a* and q* move in opposite direction.
Using elementary calculus with equation (3a), we have the following proposition.
qq
or t  q
2 , the optimal teaching effort, t*, increases with the
0
weightage of discrepancy between placement and grade,  and weightage of placement, . It also
decreases with the rate of assessment disutility, . By virtue of equation (1), the same relationships
hold for e* also.
Proposition 1: For
0
We would also like to illustrate the above situation in numerical terms, This has been done in the next
section, Table 1.
The relationship of t* and e* with the (im)precision parameter of placement package, , is more
difficult to investigate analytically. The relevant derivative turns out to be


dt 1
1




d t 3 2  2q  t
 

2  

0
1
13 
 
………………….(6)
6
24 4

   
4

2
2
de dt , so if dt
0

d
d d
It is easy to establish that
then
de
 0 and de  0  dt  0.
d
d
d
We will further address this issue with help of a numerical illustration in section 3, table 2.
Case 2:
q  q or t  q
0
2
0
.
From equation (3), in this case we obtain
3 

 

4( 2) 
 
 4

1
t
4
3
1
3
2
…………………………….(7a)
3
3
2
t*
e* 

2
3 
3
4
3
4

1
……………………………(7b)
2
 

2 
 
 4

11
3
4
4
2
Here, for a solution we need


4
(otherwise a solution will not exist). Note that both t* and e*
2
decreases as  (weightage on placement performance) increase.
Also, the assessment effort is given by
a* 
3


2
t *.
So, for the low quality case, a* and
t* increase together. The comparative static results with respect different parameters will also be
similar.
5
In this case, we have the following results.
Proposition 2: For
qq
0
or
tq
2
0
, the optimal teaching effort, t*, increases with the
(im)precision parameter of placement package, , weightage of discrepancy between placement and
grade,  and rate of assessment disutility, . It also decreases with the weightage of placement, . By
virtue of equation (1), the same relationships hold for e* also for ,  and . For , a bit of tedious
calculus also yields the same conclusion.
The reaction functions (1), (2) and the quality output function, q, are shown in the following figure. It
illustrates the two alternative equilibria according to cases 1 and 2 as discussed above. One is the high
type equilibria q  q or t  q
2 (with a correspondingly high e). The other one being
0
qq
0
or t  q
0
represented in the figure below.
the low type, with
0
2 (correspondingly low e). Both are qualitatively
Figure 1: Schematic diagram for equilibrium configurations of t* and e*
e
q=
te
R (t ) : e 
s
t
2
R (e)
t
q=
te
q
0
t
2
Again, refer to table 1 below for the numerical illustration.
In terms of assessment efforts, we have the following summary for the two cases discussed above.
Proposition 3: (a) For high quality output case ( q
q
), t* and a* move in opposite direction.
0
High t* implies low a* (high t*  high e* also and hence high quality but in this case a* is chosen to
be low).
(b) For low quality output case ( q  q ), t* and a* move together (low t*  low e* also  low
0
q* and in this case a* is also chosen to be low).
As assessment imprecision is highest for middle values of q*, this result is quite intuitive. The above
proposition may be summarised in the following diagram.
6
Figure 2: schematic diagram for relationship between q* and a*
a*
q
q*
0
The Payoffs
For the above two cases, we also need to look at the value of the student’s pay off and the teacher’s
incremental pay off,  and  (say) respectively. The expressions for these are as follows:
2
t*
……………………………..(8)
* 
8

1
  
 *  
 t *  d (q*)  d (q*) 
2
a*
2
 4
2
and
2
2
3
………(9)
2
The above expressions readily yield the following conclusions.
Proposition 4 (Student’s Payoff): Student’s payoff directly improves with t* always in a quadratic
relation. (t* increases  e* increases   increases).
It is also important to study the relationship between  and .
In Case 1, it is difficult to analytically evaluate d * (likely to be < 0). See table 2 for a numerical
d
illustration.
For Case 2, As t* involves a power of  at
7
4
, that of * will be
3
2
, and hence
d *
 0 . The
d
numerical illustration in Table 2 supports our analytical findings.
3. Numerical Illustration
In the previous section, some of the important comparative static exercises were not possible to do
analytically; we take these up for investigation in numerical terms in this section. Among the
parameters of the model, we have kept the teacher’s relative preference for teaching, , and the
weightage on grading discrepancy penalty, , at constant levels. We choose  = 0.5 and  = 1. We
7
have also kept the value of
q
0
constant at 2. We then alternately vary the values of weightage on
placement, , and the (im)precision parameter, , to explore the comparative static implications.
We first fix  at 2 and vary the value of  over a few alternatives to compute the optimal values for t,
e, a and q and the resulting values for the teacher’s and student’s payoff. The values of  are chosen
so that both case 1 (high quality) and 2 (low quality) are illustrated. The results are given in the
following table.
Table 1: Results for different values of  (keeping other parameter fixed as mentioned).

2
4
6
9
10
11
12
14
16
Relevant
case
1
1
1
2
2
2
2
2
2
t
e
a
q


7.984283
7.946187
7.484894
6.44742
3.833659
2.828427
2.279507
1.681793
1.355403
1.996071
1.986547
1.871223
1.611855
0.958415
0.707107
0.569877
0.420448
0.338851
0.250492
0.37754
0.801615
1.86121
1.565085
1.414214
1.316074
1.189207
1.106682
3.992141
3.973093
3.742447
3.22371
1.916829
1.414214
1.139754
0.840896
0.677702
-8.06272
-0.2138
6.039075
11.48579
1.554684
-1.65212
-3.34784
-5.25402
-6.39342
1.992149
1.973184
1.750739
1.299038
0.459279
0.25
0.16238
0.088388
0.05741
We generate a graph between a* and q*, based on table 1, below to verify our conclusions depicted in
figure 1. It is seen that the numerical pattern follows our theoretical conclusions.
Figure 3: Change in a* with q*
a
2
1.5
1
0.5
0
0.6777 0.8409 1.1398 1.4142 1.9168 3.2237 3.7424 3.9731 3.9921
q
a
We next fix the value of  at 2 and vary , keeping the values for other parameters fixed as before.
Again, we illustrate both the cases. The computations are presented below.
For the parametric configuration of table 2,
dt
d
turns out to be positive but notice that
de
d
is
positive for values of  up to 1.25 but is negative thenceforward. This again supports our intuitive
discussion with equation (6) earlier. Also, as discussed above, note the non-monotonic relationship
8
between t* and a*illustrated both in tables 1 and 2. It is positive for the low quality cases but reversed
for high quality, verifying proposition 3.
Table 2: Results for different values of  (keeping other parameter fixed as mentioned).

0.2
0.4
0.6
0.8
1.25
1.5
1.75
2
2.25
2.5
Relevant
case
2
2
2
2
1
1
1
1
1
1
t
e
a
q


0.070293
0.26134
0.651539
1.659555
6.128098
6.882931
7.460028
7.984283
8.473203
8.934291
0.175732
0.326675
0.542949
1.037222
2.451239
2.29431
2.131437
1.996071
1.882934
1.786858
0.919639
1.140405
1.268915
1.290744
0.53753
0.33972
0.279278
0.250492
0.233987
0.223464
0.111142
0.292187
0.594771
1.311993
3.875749
3.973862
3.987553
3.992141
3.994306
3.995536
-1.20808
-1.75443
-1.93176
-1.0359
2.195342
-1.48908
-4.94396
-8.06272
-10.94
-13.6446
0.015441
0.053358
0.147397
0.537914
3.004286
2.63193
2.271511
1.992149
1.77272
1.596431
We now illustrate the relationships between student’s payoff, , given by (8) and  in the following
figure based on table 2.
Figure 4: Change of  with , numerical illustration (for  < 1 we have the high quality
case 1 and  > 1 implies the low quality case 2)
For teacher’s payoff, a direct computation of
d
d
was infeasible in either case because of the
complicated algebraic form, so we take the help of table 2 for a numerical illustration in the following
Figure.
It is seen that the student’s and teacher’s payoff increases with  when  is small. This happens
because with an initially high precision (low ), a marginal increase in it leads to a higher effort
allocation by the teacher in teaching, leading to equivalent increase in e*. As a result, it leads to a
higher quality output which gets rewarded in the job market (as  is still low). Once  becomes large,
then this award no longer happens as the resulting screening mechanism is no longer able to
9
distinguish quality to a considerable extent. Thus both  and  starts falling. As a reaction, the
student starts choosing lower e* resulting in a subsequent slow increment in quality (see table 2).
Figure 5: Change of  with , numerical illustration
Choice of  and 
To generate high quality output, it is imperative to create incentives for the teacher and student to put
in a strong effort. From the teacher’s point of view, a moderate value of  (in the range of 6 to 9 in our
illustration) and a value of  slightly higher than 1 (around 1.25 as in the illustration above) seems to
be beneficial. This means that a very high or very low weightage on placement performance does not
create strong incentives for the teacher. The student would of course prefer a very low value of , but
even a moderate value will generate a fair incentive structure for the student also. Notice that such
choices of parameter values will also lead to high quality output and low discrepancy. This has the
additional advantage that a lower level effort need be spent on assessment.
4. Conclusion
Teacher's allocation of effort between teaching and assessment and its consequences on human capital
formation has caught the attention of academic researchers for some time. This paper develops a
model in a principal agent framework to study the interaction between the governing body, the teacher
and the student of an academic institution. The choices of unobservable effort levels by the teacher
and the student for the relevant activities can be manipulated by the governing body by choosing
appropriate incentive mechanisms. In a sequential move game, we solve for all such mechanisms by
the method of backward induction and establish necessary and sufficient conditions for these
mechanisms to be equilibrium behaviour under alternative payoff structures. The trade-off between
teaching and assessment effort is sharply delineated in this model.
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