This paper studies identification and estimation of average causal effects, such as average marginal or treatment effects, in fixed effects logit models with short panels. Relating the identified set of these effects to an extremal moment problem, we first show how to obtain sharp bounds on such effects simply, without any optimization. We also consider even simpler outer bounds, which, contrary to the sharp bounds, do not require any first-step nonparametric estimators. We build confidence intervals based on these two approaches and show their asymptotic validity. Monte Carlo simulations suggest that both approaches work well in practice, the second being typically competitive in terms of interval length. Finally, we show that our method is also useful to measure treatment effect heterogeneity.