Determining the effect of adding an independent variable by comparing the coefficients of different regression models with subset of Xs? I often see this done in multiple regression modelling but I am not sure if it is correct:
Suppose one has three explanatory variables, X1, X2, X3, and a dependent variable Y. (Assume all variables are continuous)
One creates several regression models with subsets of this Xs'as explanatory variables. For instance,
(a) Y=Bo+B1*X1+B2*X2+B3*X3
(b) Y=Bo+B1*X1+B2*X2
(c) Y=Bo+B1*X1+B3*X3
Understandably, the values of the coefficients will differ if you add/ remove an explanatory variable. If we consider models (a) and (c) for instance, values of B1 and B3 in (c) will definitely be different from the values of B1 and B3 in (a). 
Let's say that B1 in (a) is 0.3, and B1 in (c) 0.5. Is it correct to conclude that removing X2 changes the effect of X1 on Y? i.e. can we say that the effect of X1 on Y will increase if X2 is not included?
Similarly, if we add another variable X4 to model (a), creating Y=Bo+B1*X1+B2*X2+B3*X3+B4*X4, and the coefficient of B1 in this model becomes 0.7 (considering the same example that B1 in (a) is 0.3), can it be said that X1 will have a larger effect on Y if X4 is present?
Are these models directly comparable?
I hope my question is clear. 
Thank you for any help. 
 A: 
Understandably, the values of the coefficients will differ if you add/
  remove an explanatory variable. If we consider models (a) and (c) for
  instance, values of B1 and B3 in (c) will definitely be different from
  the values of B1 and B3 in (a). Let's say that B1 in (a) is 0.3, and
  B1 in (c) 0.5. Is it correct to conclude that removing X2 changes the
  effect of X1 on Y?

It depends on what you mean by 'effect' -- this is phrased as if it's a causal question, but I can't tell whether that's what was intended.
Strictly speaking, the effect of X1 on Y is whatever it is; it doesn't depend on what variables are in the model.  B1 in (a) and B1 in (b) are different, so at least one of them is not the effect of X1 on Y. It's quite likely that neither of them is the effect of X1 on Y.  There are known criteria for deciding when a regression coefficient estimates the effect of a variable; they are complicated and use, for example, causal graphs.

...can it be said that X1 will have a larger effect on Y if X4 is
  present?

Again, this is phrased as if it's a causal question. The question makes sense, if your are talking about effect modification.  The effect of changing X1 may depend on the value of X4: the effect of crossing the street without looking depends on how much traffic there is.  To start to address these sorts of questions you need models for interactions.
A: Equation of Regression will get a composite cofficient when absent of a variable.
Let me intepret for your case 
(a) Y=Bo+B1*X1+B2*X2+B3*X3 with
   B1 = 0.3 
   B2 = 0.6
(c) Y=Bo+B1*X1+B3*X3 with
   B1 = 0.5
Assuming that i have a regression between X2 and X1
(1) X2 = Ao + A21 * X1 
    with A21 = 0.33333
Now let see again (a) with absent of X3:(here i only focus on changing coeficient of regression of X1)
(a) =>Y = Bo + B1*X1 +B2(Ao + A21 * X1) +B3 * X3
=>Y = Bo + B1*X1 +B2 *Ao + B2 * A21 * X1 +B3 * X3
=>Y = (Bo + B2 *Ao) + (B1 + B2 * A21) * X1 +B3 * X3
Let see a new B1* = 0.3 + 0.6*0.33333 = 0.5
likewise in case absence of X4 with B4*A41 = -0.4
in persionally, i think you need consider to another variate (R^2 or Adjusted R^2) to identify whether what model is more approriate.
A: First of all, you should always fit a univariate model, i.e. Y~X1, Y~X2 and Y~X3 to understand the direct effect of the each predictor on the response variable. Only if the effects are significant, then we consider multivariate regression models, adding the predictors one at a time into the model to try and obtain the "best fit models" via methods such as stepwise regression.
Assuming that X1, X2, X3 and X4 are all significantly associated with Y, when comparing coefficients of the betas in (a) and (c), you can say that when you further adjust for X2 in the model, the effect of X1 on Y becomes attenuated. Usually, if adjusting for a predictor in the model changes the beta coefficients significantly, it would suggest that the variable is confounding the relationship between Y and the predictors. In your case, adjusting for X2 would provide a more accurate estimate of the effect of X1 on Y which is 0.3. 
You should also check to see the cross-correlation matrix of the predictors to make sure that there is no highly correlated variables. If there is, you should not put them into the same model to avoid collinearity issues which could be why your coefficients are changing so drastically.
