How much effort to put into a study assignment. A basic model

Table Of Contents

Introduction

The model

First extension

Second extension

Policy implications

Conclusion

Further Extensions

Introduction

Every semester millions of students worldwide have to choose their courses, read compulsory literature, intensify their knowledge in their chosen field of studies and pass exams to advance further in their academic life. Students prepare themselves for their future jobs, improving skills in different areas, most of which will be applicable in specific business environments. They study to have a better perspective in the nowadays tight job market, to increase their expected future income, as they signal their quality as an employee by their degrees and grades. Where the potential future job however offers the paid wage as an incentive, for every hour or month worked,, your studies lack these direct monetary incentives. Being the better student will not yield a higher pay directly, you will not get money for every passed exam or every top grade you get in your studies.

The question arises then: How do students decide on how much effort to put into their studies? Do students always aim for the best grade possible? What are the influencing factors students take into account when they face the decision on how much time to spend studying, trying to improve their grade in a given assignment they have to hand in? Do students wake up in the morning and think about their wage in 3 years, leaving for the library at 8 a.m. to improve it? What other parameters might alter their decision to do so? This paper sets up a basic model to explain the influence of different factors in a students’ decision process of how much effort to put into an assignment.

In the first part the basic model is set up, explained and the first conclusion, how a typical student behaves optimally, is derived. The second part establishes the first extension of the model, by introducing a competition parameter between students. It will be analyzed what consequences this competition has on social welfare and which problems occur. The third part lifts the “typical and identical” student assumption and introduces different types of students. It is then derived, if and when different types of students should study and when they should not. In the fourth part policy implications and solutions to the problems out of part two are discussed. The paper finishes with a conclusion and an outlook on possible further extensions.

The model

In the first period of this two period model, the student decides on the level of effort she puts into studying for the assignment. This level of effort is denoted by s, which can be interpreted as the hours the student spends in the library, working on the assignment. It is assumed, that an additional hour of studies will increase the final grade of the student, i.e. there is no “idle” time studying, where the student does not make progress. However, the grade does not increase linearly in s, as studying exhibits positive but diminishing returns to scale. The direct utility the student gets from the final grade is given by and represents the mere satisfaction of obtaining the grade in monetary value, where λ is a scaling parameter to convert the utility to monetary units. To put it a different way, can be interpreted as the money the student would be willing to pay for the grade he gets.

The parameter α represents the student’s ability to turn his effort effectively into a better grade. Further, the final grade will influence the student’s future wage Π. The magnitude the assignment’s grade will influence the wage is given by ε<1. This ε can be varied, as the part of the assignment in the total GPA of the student varies. As this model is not restricted to calculate the optimal effort level of a small assignment but could also be used to determine the optimal effort level of a whole degree, ε can be changed accordingly. For a single assignment ε would be assumed to be close to zero, as one assignment does not influence the GPA and therefore the student’s future wage in a drastic way. The future wage, being paid in the second period of the model, is discounted by the students discount rate (1+ρ). The factor ρ does not necessarily have to be equal to the real interest rate (r), as usually students do not think explicitly about their future wage, when they write an assignment.[1] This way, it can be accounted for a different view of the student between monetary payments which are a direct result out of the studies, like grants, and the more indirect monetary payments, the future wage.

Finally, as studying is work most of the time and students enjoy leisure more than work, studying incurs a cost c per hour of work. This is a fetch variable, which includes everything from the foregone wage the student could have earned working, to the utility she would get out of spending the hour outside in the park, drinking coffee, or leisure in general.

Why do the benefits of effort exhibit positive & diminishing returns to scale?

While the first hour a student puts into learning or working on an assignment yields very large gains in understanding and progress, the 100th hour usually still adds some more understanding and progress, but far less than the first one. Imagine a student who starts learning for an exam. Sometimes the first day of learning will already suffice to merely pass the exam. If the student wants to get a good grade however, she has to invest more time. To get a “7,5” may take a week, as she has to be prepared for all the different types of exercises, which can be asked. To get a “10”, if even possible, will take even more time. The student has to be perfectly prepared on every single task which can be asked of him, has to be able to connect all the single dots of the course to see and understand the bigger picture. The same is true for an assignment: To put together an assignment which will merely pass may take only two or three days, even less, depending on the length. To score a higher grade, the assignment has to be more elaborate, all the details have to be in the right place and connected to each other. If one thinks of an assignment, which would be graded “8” and is mostly finished by the student, the amount of additional time she would have to spend to increase the grade to a “10” is much higher than for a student who currently works on an assignment which would be graded “4” and she wants to increase her grade to “6”.

Why are the costs linear?

The cost function is assumed to be linear for simplicity. However, this is not an unrealistic assumption: Every hour the students spends on the assignment yields the same opportunity costs, for every hour he works he loses one hour of leisure. One could argue that people get fed up with a single task if they have to spend an enormous amount of time on it. At one point the student just wants to get it over with, which would mean marginal costs would be positive and rising in effort. On the other hand, the more knowledge one gains about a specific topic, the more interesting it gets. The more insight one gains, the easier it is to work out a different perspective, a new angle on how to approach the problem. This would be an argument for positive but decreasing marginal costs in effort. Even though one can argue for many different shapes of the cost function, in this basic model, it is assumed that both effects even each other out and the costs are linearly increasing in effort.

The optimization problem the student faces then is:

Solving for s leads to an optimal effort level of:

For simplicity, λ is assumed to be normalized to one but could be introduced as larger zero to scale the direct utility of the grade up or down in relation to the future wage.

The optimal effort level rises in α, the future wage Π and the magnitude parameter relating the grade to future wage ε. It falls in the cost of studying c and the students discount factor (1+ρ). The intuition behind these effects all are straight forward: If the student has a higher return in grade-terms from studying, he will increase his effort. If his future base wage increases, which he can influence by effort, he will put more effort in his studies, as well as if his direct influence ε on the future wage increases. Higher costs of studying naturally decrease the optimal effort and if the students discount factor increases, the student cares more for todays leisure time, than for tomorrows wage, lowering his optimal effort level today as well. As the model is not constrained to a fixed time period to account for different lengths of assignments or even degrees, it is important to look at the derived optimal effort level: For every modeled study-activity, the optimal level has to be measured against a specific time constraint, depending on the size of the activity. If the assignment for example has to be handed in within two weeks, a possible time constraint would be 196 hours: Assumed the student cannot work more than 14 hours a day on her studies, the maximum amount of time she can study in 14 days are 196 hours. If is larger than 196 for this assignment, the optimal level is a corner solution and the student should do nothing else than studying in these two weeks.

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