A formalization of negelectness and it’s implications

Assume that doing some area of doing good has a production function 

ui =A(f(x1,x2...xn) where x is a input like talent, money, political capital, ideas ect and A is a measure of productivity. 

We could also have some utility function U(u1,u2,... un) where ui is a quantity of some thing like like level of carbon dioxide in the atmosphere, number of animals in factory farms, number of children under 5 dying ect. 

The existing EA view of neglectedness I think is that the less resources a field has going into it the more worthwhile it is to work in the field because there are diminishing marginal returns to inputs but broadly constant returns to U from each u. 

I.e  partial derivative G/xi <0, partial derivative U/ui = some constant k.

I think this is probably the wrong view. 

The conceptual case

1.Within economics normally A is viewed as the technology parameter, and increasing A increases the productivity of other inputs like money and talent. Within a good context this could be EA ideas. For instance within climate change applying EA ideas could dramatically increase the productivity of the money and talent already in the field. In this case the diminishing returns that would be faced would be on the utility function side in which, for instance, reducing the first 10% of carbon reduces x-risk from climate change by 50% or something 

2. It may well at large levels be reasonable to that the production function for ui displays constant or diminishing returns to scale, implying that  G/xi <0. However, this could still imply that it’s worth putting resources into a specific xi because this will have effects proportional to the amount of resources already in the production function and the ‘technology’ being used by the production function. This implies that all else equal the most resources should go to where there’s the greatest difference between inputs. For instance if a field has lots of money and not much talent this shouldn’t be lumped together into ‘this field is somewhat neglected’, it implies that putting in talent should be prioritised.

3.It may not be reasonable to assume that the production function does exhibit constant or diminishing returns to scale. For instance because the more people working in a field the more they can specialise, or because more phd programs get set up focusing on the field in particular. This could be non-linear, for instance it maybe that only past some threshold does the probability of universities having undergrad modules on a topic increase. 

4.There may be increasing returns within the utility function. For instance, the lower the rate of x-risk, the more valuable it is to reduce x-risk because the expected value of the future is greater. Potentially this could also be the case in education in low income countries. If 10% of a town is able to read no one sets up a garment manufacturing centre, but between 25% and 75% they become likely to, with declining returns after 75% (to be clear, I just made those numbers up.) 

A numerical example.

u1=malaria nets

u2=carbon dioxide emissions

u=A(L^(a)K^(1-a))

Malaria nets starts off with 100 units of capital, 10 units of labour, a=0.5, A=100

Carbon dioxide emissions starts with 30 units of capital, 30 units of labour, a=0.5, A=100

u1=3162

u2=3000

So in this case carbon dioxide emissions have fewer total resources allocated to them and are producing fewer units of output (assume for simplicity that we have the same utility functions over one unit of carbon dioxide emissions and 1 unit of malaria nets.)

When deciding where to put resources next, assuming we have a single shot of additional resources we should compare the partial derivatives, ui/L, ui/K. In this case u1/L is the highest at 158.