Deep learning models that perform well often have high computational costs. In this paper, we combine two approaches that try to reduce the computational cost while keeping the model performance high: pruning and early exit networks. We evaluate two approaches of pruning early exit networks: (1) pruning the entire network at once, (2) pruning the base network and additional linear classifiers in an ordered fashion. Experimental results show that pruning the entire network at once is a better strategy in general. However, at high accuracy rates, the two approaches have a similar performance, which implies that the processes of pruning and early exit can be separated without loss of optimality.