A researcher fits a simple linear regression model: $\hat{y} = \beta_0 + \beta_1 x$. To make the intercept more meaningful, she decides to center the predictor variable by subtracting its mean, defining $x' = x - \bar{x}$, and fits the model $\hat{y} = \alpha_0 + \alpha_1 x'$. What is the relationship between the new intercept $\alpha_0$ and the original variables?