I will try to explain my question using two production functions here.
$Y$ = Yield of a certain crop (tons/hectare)
Assume yield (output) is a function of two inputs, $Y = f(N,I)$, where
$N$ = Nitrogen fertilizer (input) (kg/hectare)
$I$ = Irrigation (mm/hectare)
Case 1: Assume I have data on 400 different plots. Each observation includes yield (output) and N (input). There is no irrigation in this case, $I = 0$.
$Y_1$ = $\alpha_0$ + $\alpha_1$ * $N_1$ + $\alpha_2$*$N^2$ + $e$
Case 2: Irrigation (say up to 100 mm/hectare) is introduced in all plots, therefore, the value of $I$ ranges from 0-100. Irrigation increases yields in most plots. However, for a large number of plots, irrigation alone does not increase yields and additional N is required to increase yields. I have data on both inputs and the corresponding yields (output).
$Y_1$ = $\alpha_0$ + $\alpha_1$ * $N_1$ + $\alpha_2$*$N^2$ + $\alpha_3$ * $I_1$ + $\alpha_4$ * $I^2$+ $\alpha_5$ * $N*I$ + $e$
What I want to measure is: (1) What is the impact of irrigation on yields? (2) Given that irrigation increases yields, how much additional Nitrogen (on average) is necessary to get to these yield increases?
a) From what I understand, I can't use a diff-in-diff type estimation here because the impacts of irrigation are heterogeneous, i.e., I can't simply add an irrigation dummy to the regression.
I would appreciate any help. Thank you so much!