How to get the effect of one dummy variable against many others? I have the following regression:

wage = constant + (beta1)*michigan+ (beta2)*california+...+(beta49)florida+(beta50)education + u

where michigan is equal to one if the person is from michigan and zero otherwise, and so on for all of the next state variables.
education is equal to the number of years of education of the person.
The base case is South Dakota, which is the state that does not appear in the regression, that is why the coefficient for florida is 49 and not 50.

I want to get the effect of being from south dakota against any other state in wage. My idea is to run this regression, and then add up betas for all other states (beta1 to beta49). If the sum is positive that would mean that people from any other state earn on average more, and the opposite if the sum is negative. Is this correct? if so , why? if not, why not? and what would be the right way to do it?
If you only want to test whether the mean wages of South Dakotans is higher than for residents of other states, run:



Where South Dakota is a dummy variable like the one you've described earlier. I'm assuming you simply have a cross-section of wages at a single point in time, but if you have either repeat observations of individuals or a repeated cross-section, then you can include a t subscript for each of the variables and include time controls.  

If  is positive and significant, it would be interpretable as the level or percent (if the model is run in log wages) increase in mean wages for a resident of South Dakota compared to the mean for the pool of residents of all other states. 

If instead you wanted to perform pair-wise comparisons between South Dakotans and residents of an individual state, you could include a dummy for that state and then run a command like lincom in Stata to perform hypothesis tests between those states' coefficients.   

If you wanted to compare South Dakota against every single other state, then make South Dakota the omitted category in your regression, and you could run the model you described of including dummies for all non-South Dakota states. The big issue with this approach is the massive multiple hypothesis testing issue you will encounter, which could be addressed through Bonferroni correction.