Self-distillation (SD) is the process of first training a \enquote{teacher} model and then using its predictions to train a \enquote{student} model with the \textit{same} architecture. Specifically, the student's objective function is $\big(ξ*\ell(\text{teacher's predictions}, \text{ student's predictions}) + (1-ξ)*\ell(\text{given labels}, \text{ student's predictions})\big)$, where $\ell$ is some loss function and $ξ$ is some parameter $\in [0,1]$. Empirically, SD has been observed to provide performance gains in several settings. In this paper, we theoretically characterize the effect of SD in two supervised learning problems with \textit{noisy labels}. We first analyze SD for regularized linear regression and show that in the high label noise regime, the optimal value of $ξ$ that minimizes the expected error in estimating the ground truth parameter is surprisingly greater than 1. Empirically, we show that $ξ> 1$ works better than $ξ\leq 1$ even with the cross-entropy loss for several classification datasets when 50\% or 30\% of the labels are corrupted. Further, we quantify when optimal SD is better than optimal regularization. Next, we analyze SD in the case of logistic regression for binary classification with random label corruption and quantify the range of label corruption in which the student outperforms the teacher in terms of accuracy. To our knowledge, this is the first result of its kind for the cross-entropy loss.