I would build a bivariate response model simultaneously for $Y_{i1}$ and $Y_{i2}$, introducing an underlying propensity $Y^*_{i1}$, $Y^*_{i2}$ such that
$$Y_{i1}=\max(a,Y^*_{i1}), Y_{i2} = \min(b,Y^*_{i2}).$$ If $X_i$ is a treatment variable, then a version of your model is $$Y^*_{i2} = Y^*_{i1} + \beta_0 + \beta_1 X_i + \epsilon_i.$$

I think you use Stata, so you would want to look at `cmp` <a href="http://ideas.repec.org/a/tsj/stataj/v11y2011i2p159-206.html">package</a> that fits models of this kind. There might be some caveats, as Roodman talks about "fully observed" models there, in the sense that nothing depends on the latent variables $Y^*_1, Y^*_2$. You might have to change somewhat the model to make sure it is estimable.