Why is $E(\hat{\beta}_1) = \beta_1 + 1$ also an unbiased estimator of true $\beta_1$

Question

let regression model be $$Y_i = \beta_0 + \beta_1X_i +u_i$$

If the ordinary least square estimator of $\beta_1$, $\hat{\beta}1 = \frac{\sum^n_{i=1}(x_i -\bar{X})(y_i-\bar{Y})}{\sum^n_{i=1}(x_i -\bar{X})^2}$ is an unbiased estimator for true $\beta_1$. Is $\hat{\beta_1}$ also an unbiased estimator of the true $\beta_1 + 1$? In other words, is $E(\hat{\beta}_1) = \beta_1 + 1?$

Question: Why would $E(\hat{\beta_1}) = \beta_1 + 1$ still be an unbiased estimator?

Answer 1

No

$$\mathbb E[\hat\beta_1]=\beta_1\ne \beta_1+1$$

But $\hat\beta_1+1$ will be unbiased for $\beta_1+1$.

$$ \mathbb E[\hat\beta_1 + 1] = \mathbb E[\hat\beta_1] + \mathbb E[1] = \beta_1 + 1 $$