Is there a difference between controlling for population size directly vs. putting variables in per capita terms? I want to look at the impact of a number of factors (e.g nutrition, alcohol sales, GDP, etc.) on mortality. I have a dataset with 10-30 years of data on these variables aggregated at the state level. (i.e. I'm looking at the impact of total state-wide alcohol sales in a given year on total deaths in that state in that year). Obviously, population size is a likely omitted variable if I don't include it. However, I am wondering whether I should include it by transforming variables to their per-capita form or whether I should control for population directly.
It seems that creating per capita variables is the conventional thing to do, but it seems to me that just adding in population size as a control is simpler and it allows for population size to have its own unique effect on the dependent variable. For example, in this case, we might think that states with larger populations, all else being equal, have more deaths because it is harder to organize the healthcare system efficiently to meet the demands of a large population.
What should I do here? Are these two approaches functionally equivalent?
 A: Its not that simple, at least not in a linear additive model. It also fundamentally depends on your dependent variable. 
Take a simple example where the true Data Generating Process (DGP) a increase from 1 in a 1000 individuals with a certain condition to 2 in a 1000 is associated with a 10% increase in some variable of interest and there is no association between population size proper and your dependent variable. Take two states with resp. 10,000 and 100,000 inhabitants and resp. 20 and 40 individuals with a certain treatment.  You propose adding as regressors [20, 10,000] and [40, 100,000] whereas per capita regressors would have been 2 and 0.4. Your model will not accurately reflect the association you are looking for because the per capita ratio between the two observations is decreasing (2 versus 0.4) whereas in absolute terms it is increasing (20 versus 40). Therefore, the two approaches are clearly not equivalent. Which you prefer depends on your dependent variable and the relationship you are testing, but in an OLS framework variables are independent so adding the total population as one regressor doesn't mean the other regressor is scaled by such a variable - they are approached as independent numbers by the model. Note that if your dependent variable is a rate or per capita value, you generally would want your regressors to be the same (just think about the above situation).
There is no problem, however, adding both per capita ratios AND some population size indicator (probably best to take the natural log of the true population) - that seems to be the best approach in your case?
Hope this helps 
A: Let $Y_{ij}$ be the number of death at state $i$ in year $j$, $P_{ij}$ be the corresponding # of people, $X_{ij}$ be a vector of other covariates, such as GDP.
Then fit a Poisson regression model with random state specific intercept:
$$Y_{ij} \sim  Poisson(\lambda_{ij}P_{ij})$$
$$\lambda_{ij} = exp(X_{ij}\beta + \epsilon_i)$$
$$\epsilon_{ij}\sim N(0,\sigma)$$
In most software, $\log(P_{ij})$ is called as offset.
A: I answered your more general question, Controlling for population size, using per capita or including a variable for population size with reference to this one. To address this more specific question:
No, they are not functionally equivalent. Controlling for alcohol sales per capita is controlling for sales rate. It is density of how much alcohol is being imbibed (or purchased for some other reason..?) per individual person.
If you control separately for alcohol sales and population, your regression is modeling, separately, the impact of alcohol sales AND the impact of population WITHOUT considering that the relation between the two is key.
If you control for alcohol sales per capita, your regression is modeling how much alcohol is purchased per person WITHOUT considering how much alcohol was purchased OR how many persons there are.
In short, when you say

Obviously, population size is a likely omitted variable if I don't include it. However, I am wondering whether I should include it by transforming variables to their per-capita form or whether I should control for population directly.

I think that the core of the misunderstanding becomes clear. The answer is that you can't "include it by transforming variables to their per-capita form," because per capita is not a form of the variables. It is created by the variables, but the information gleaned from the variables separately in a regression analysis is NOT simply absorbed into the per capita variable.
