Invariance of average output given output maximization Assume that here are two areas $a = {1,2}$ and that $(e_{i1},e_{i2})$ is IID Gumbel location 0 scale 1 for all $i$. Assume further that
$$w_{i1} = \mu_1 + e_{i1} \\
w_{i2} = \mu_2 + e_{i2}$$
and that individuals maximize output selecting the area $A(i)=a$ where $$A(i) = \arg \max_{j} \{w_{ij}\}.$$
What is the bias of the estimate
$$ \frac{1}{N_1}\sum_{i\lvert A(i)=1}w_i - \frac{1}{N_2}\sum_{i\lvert A(i)=2}w_i$$
of the average treatment effect of an individual moving from area 2 to area 1?
 A: It follows from the invariance of achieved utility property that the bias is equal to the causal effect with opposite sign.
The expectation of the utility achieved due to maximizing behavior is the same across alternatives (here areas or groups). This creates an endogenous dummy variable problem and the regression
$$W_i = \beta_0 + \beta_1 D_i + u_i,$$
results in inconsistent $\hat \beta_1$ with probability limit $0$. So causal effect is estimated to zero.
This is easily illustrated in a small simulation in R
library(evd)
mu_1 <- 1
mu_2 <- 2

N <- 100000
Z <- matrix(rgumbel(2*N),nrow=2)
W <- Z + c(mu_1,mu_2)
index1 <- W[1,]>W[2,]
index2 <- W[2,]>W[1,]

mean(W[1,index1])
mean(W[2,index2])
0.5772 + log(sum(exp(c(mu_1,mu_2))))    

D <- as.numeric(index1)
Y <- W[1,]*D + W[2,]*(1-D)
summary(lm(Y~D))



Residuals:
    Min      1Q  Median      3Q     Max 
-3.1663 -0.9013 -0.2118  0.6754 11.6214 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.886916   0.004731 610.251   <2e-16 ***
D           -0.003468   0.009074  -0.382    0.702    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.277 on 99998 degrees of freedom
Multiple R-squared:  1.461e-06, Adjusted R-squared:  -8.539e-06 
F-statistic: 0.1461 on 1 and 99998 DF,  p-value: 0.7023

