Clear out a textbox after submitting its content
Since you're already keeping track of the &lt;input&gt;'s value in your model, it would be cleaner to use that as the value in your view, too.

view : Model -&gt; Html Msg
view model =
    div []
        [ div [] (List.map viewMessage model.messages)
        , input [ onInput Input, value model.value ] []
        , button [ onClick Send ] [ text "Send" ]
        ]


This way, your view is a pure representation of your model, and your model is kept in sync with user input.

I took the liberty of doing a quick pass over your code, additionally introducing record update syntax in your update function, as opposed to constructing a fresh record from scratch.

Adding or removing a field from a record needn't result in refactoring every single place you happen to modify the values of that record.