I've got a problem that seems to be exposing a fairly fundamental hole in my econometrics training. I'm looking for a canonical reference for how to deal with the following sort of problem:
For example, say that firms have demand for security, and have limited budgets. They can either hire security guards, or spend money on fancy alarm systems. These goods are both substitutes and complements.
$$ Security = \beta_1 Guards + \beta_2 Alarms + \beta_3 Alarms \times Guards $$ with all three terms positive, and "security" being unobservable.
Now, I'm interested in the effect of $\mathbf{X}$ (i.e.: covariates) on demand for security, but I only measure guards and alarms. I can estimate
$$ Guards = \gamma_1Alarms + \mathbf{X\Gamma} + \epsilon_1\\ Alarms = \theta_1Guards + \mathbf{X\Theta} + \epsilon_2\\ $$
only if I have instruments for each -- things that affect demand for one but not the other. This might be possible in the example, but generally might not be. And in any case I'm not really interested in ceteris paribus relationships -- what I really want is the causal relationship:
$$ \partial\mathbf{X} \rightarrow \partial Alarms, \partial Guards $$
So, for a given change in $\mathbf{X}$, how would demand for alarms AND guards change? (And, how might their ratio change as well, but I imagine that'd be harder).
I'm sure that there must be standard methods here, as this must be a common problem. But I'm not sure what exactly to look up.
