# Paper where an integral in the constraint of an optimization problem is treated as infinite sum

I am looking for a paper (or textbook, or even lecture notes example) where there is a problem such as $$\max f(x) \\ \text{ s.t. } \int g(x) \leq \int c$$ and there is at least some exposition/explanation about solving it. I believe that these are sometimes (often? always?) treated as infinite sum, but I may be wrong.

In words, I am looking for an example and solution (or at least a discussion of solution) of a problem where the constraint contains an integral. It is okay (and probably will be the case) that $f(x)$ contains an integral too.

I believe such optimization problems occur in macroeconomics. I'm specifically thinking of integrals over possible states, but the integral doesn't necessarily have to be over states.

Or, if someone could explain solving such problems to me, that would be okay as well.

Thanks.

• It is difficult to answer because your notation is unclear. I assume the integral is to taken w.r.t. $x$, but where does the function $g$ come from? Are we supposed to maximize $f(x)$ w.r.t $f()$ or $x$? Perhaps it should be $$\max_f \int f(x) dx$$ $$s.t. \int \frac{f(x)}{(1+r)^x} dx \leq c$$ or something similar? Jan 30 '16 at 23:08

Some papers that come to my mind are ;

-Calvo and Obstfeld (1988, Econometrica)

-Endres et al (2014, Resource and Energy Economics)

-Traeger et al. (2012, European Economic Review)

In these papers, the objective function has two integrals but as these models are treating an overlapping generations model, there is a such aggregation that one of the integrals disappears because it is not possible to have an integral on an Hamiltonian. Otherwise, it is not possible to solve it by standard methods.

I am not sure about but this so take it with precaution, this kind of problem with two integrals on the objective function could not be solved by optimal control techniques but by calculus of variation.

There is a very good section about constraints with integrals on textbooks of

-Kamien and Schwarz (Dynamic Optimization : The Calculus of Variations and Optimal Control in Economics and Management)

-A. Seierstad and K. Sydsæter (Optimal Control Theory with Economic Applications)

• Thank you. In searching for a method I came across the calculus of variation, but have not had time to look into it. Jan 31 '16 at 0:40