I cooked up this topic on my way to work in the subway. My colleagues and I often discuss things like how much time should I spend on a grant? How many grants should I write per year? Is there an optimal time commitment? How much is my time worth? I decided to do this math a bit more rigorously and see what happens. I assume a working knowledge of high school AP calculus, except for the multiple grants section—talk to me directly if you want to work on that. 

A cursory search does not reveal a lot on this topic in the econ lit (JSTOR/Google scholar)...

Getting a Single Grant

Let’s say the probability of getting a grant is p, and the value of a grant is V, and the time it takes to get the grant is t. The expected value of getting the grant E=pV and per unit time is pV/t. Very straightforward. The back of the envelop calculation for a NIH research project grant is 250k per year, and let’s say you realistically spend 10 hours on writing a R01 per year, making a very conservative assumption of funding rate of 10%, the per hour expected value of money raised is $2500. This shows that fundraising is probably the most valuable activity on a dollar and cents basis, if you can limit the number of hours you spent doing it.

Indeed, increasing the number of hours for a single grant is very inefficient:

Source: sean luo


We see that the decrease of expected value goes as a square of time spent on the grant, rather than the grant itself. 

However, many might object, what if in order to make a grant better, we need to spend more time on it? Fair argument: p is a function of time. We can make a very easy model of this relationship, as whether you get a grant is a binary outcome, p can be simply a logistic function, and plug that into the expected value function and take the derivative:

sean luo
Source: sean luo


You see that it still generally goes in negative squared of t. The movement of the logistic function is fairly straight in the middle, which is where most of us stand (flip a coin):

Sean Luo
Figure 1
Source: Sean Luo

I got this function by playing around with parameters, but the units are about right: y is the p value and t is number hours. You might object to this function that p seems too high at a maximal number of hours, but remember we are not modeling other parts of p such as the “innovativeness” of the grant and other intrinsic qualities to the science of the grant unrelated to how much time you spend on it.

Plotting the expected value function by unit time now:

Source: SEAN LUO

Fig 2
Source: SEAN LUO

We see that there’s some optimal number of hours one should spend on a grant opt(t) that is somewhere between 10 and 15. Let’s try to solve it:

Source: SEAN LUO

Wolfram Alpha tells me:

Source: SEAN LUO


where Wn is the “analytic continuation log product function”. Whatever that means... irrelevant. What is relevant is that note (if you can read it) opt(t) is unrelated to the value of the grant! I.e. the optimal time you should spend on a grant if you are only considering the grant itself, should not be influenced by the value of the grant, but only on the ways in which your effort translates to increased probability of funding (i.e. coefficient k) and the minimal number of hours a grant requires (t0)

This of course assumes that V itself is unrelated to t: i.e. the size of the grant does not relate to how much effort it takes to write it. This is often the case in my experience, but we should let data speak. Roughly page number correspond to hours needed. NARSAD = 2 pages, R23 = 6 pages, R01 = 12 pages:

Sean Luo
Source: Sean Luo

Looks like it's more like a square and not quite an exponential... I plotted a few of these functions (not shown here). The intuition is the following: if grant values don't depend on time spent, after a certain amount of time spent, your per hour return actually drops. (fig 2) If your grant value scales linearly with time (i.e. just writing one R01 after another), there is a flattening of value per unit time after a certain "minimal acceptable effort" (fig 1). If your grant value scales in polynomial, your time's value spent actually increases per unit time devoted to writing grants (plot not shown, just trust me)! Unfortunately, the scaling of value by time doesn't go up by square (i.e. I can't write a 24 page grant and get 4 millions...), and at some point it's all linear value to time scaling. 

Conclusion: write R01s if you can. If you can write large U or P grants, write those. 

Multiple Grants

So why is our intuition such that opt(t) should be affected in complex ways by the value of the grants? This only happens when one writes multiple grants. The math goes something like this. Suppose one writes N grants, indexed by i. Total expected value would be:

Sum i,(Ei)

This quickly becomes complicated because one needs to do a constrained multivariate optimization where

max (Sum i,(Ei)), with constraint that Sum(i, ti) = T

T is the total time you have available to write grants.

One needs to identify an optimal time allocation scheme ti=f (Vi + other variables) such that this scheme is maximally optimal in the entire possible set of such assignment schemes.

This is where you can write a paper. Here is the outline of the paper: you can model the assignment scheme first as a linear function (i.e. ti = W V, where W is the weight matrix given for the grant value matrix), and derive a formula by solving for the matrix inverse. Then you generalize this to a specific kind of assignment (i.e. let’s say modeling f as some kind of continuous differentiable function), and then do variational calculus to solve for the form of f. Finally, you can prove some existence/asymptotic results: do you always get a unique optimal assignment, do you always get the same expected value at optimality, properties of the local extremes, etc. etc. a lot of boring details.

A Few Practical Thoughts

First, are academic researchers paid fairly? We see that per hour fund raised is about $2500-$15000. Back of the envelope calculations show that if each year one is sustained on one R01, your actual expected value of money raised is only about 250k*0.1 = 25000. Even if we use a very generous funding rate of 20% rather than 10%, you can only expect a value of 50k per year. Given the salary differential between postdoc and PI is somewhat close to that number, the “commission” given to a PI with a single R01 as a percentage is extremely high. So this exercise actually shows the reason professors are not getting paid very much is primarily due to the fact that professors (unlike investors) just can’t raise that much money, which makes a lot of sense (i.e. we don’t generate a profit---we generate knowledge that at times don’t have obvious monetary value). But this also shows that universities are in a tough spot because there’s really not much more to give if your salary is pure soft money.

I kind of want to design a calculator for fundraisers that lets you put in the dollar award, your total number of hours available to write a grant (or any kind of fund raising opportunity), and funding rate, and tell you what the expected dollar raised per hour would be, so as to let you have some sense of whether something is worth applying for or not. I bet this is also helpful to department chairs.

This by the way is not just about grant applications. This is about any kind of credit assignment issue with an intrinsic risk (i.e. the mathematics of fundraising in general). The more generalized formulation is trying to calculate the efficient frontier of allocation of time for risky return, but since the monetary value of time is not very linear, I’m simply deriving some intuition here using very simplified calculations. 

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