I am reading this paper on Economics, but I am stuck in what it seems a simple 2nd order Taylor expansion.
There are two key equations:
$$C(q,\theta)$$ $$B(q,\eta)$$
These are a cost and benefit function respectively, which depend on the quantity produced, $q$. Additionally, both functions have uncertainty, which is given by $\theta$ and $\eta$ respectively.
This is the central issue and the expansion:
I can understand most of the terms of the expansions. For instance, the first term in each equation is the value of the respective function around the reference point $\hat{q}$, as the author states:
All the other terms are consistent with an univariate Taylor expansion, except for $\alpha(\theta)$ and $\beta(\eta)$. I cannot understand where they come from.
This does not seem to be an expansion of a multivariate series, but an univariate one, as revealed by the $(q-\hat{q})$ terms. In other words, the expansion is around $\hat{q}$, and not around $(\hat{q},\hat{\theta})$. This seems to be a type of univariate Taylor expansion of a multivariate function, if that thing exists. I haven't seem such type of expansions anywhere.