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How many observations do I need?

The answer depends on the problem. For example, one observation with $X_1=0$ will give you a correct answer for $\beta_1^w$ and a standard error of zero. On the other hand, two million observations with $X_i=0$, won't be enough to estimate $B^b$. Another way to think about this question is that the basic model has five parameters and (like a regression model with five parameters) you would probably want at least 30-50 observations or so. Problems for which the parameters are more highly correlated (like regression problems with high collinearity among the explanatory variables) will require additional observations to achieve the same level of confidence. (That is, check the tomography plot to ascertain what kind of problem you are working with.) You will also want additional observations if your substantive problem demands answers with more precision. Of course, if you can selectively add observations that are especially informative, then there may be great power to be had from collecting just a bit more data.