i feel your pain.
after some messing around with siri, the music app and the podcast app. here's a solution that works (kind-of). i hate it, but i'm a subscriber of the economist too and this is the only way i know to play individual chapters.
basically if you can get music app -- note NOT the iTunes app and NOT the podcast app, but specifically your music app --, to start playing your podcast then you can see the chapter list inside that podcast.
the problem is that there is no easy way to launch your music app and point it at your podcast (how dumb is that ?). the work around i've found is to use siri. if you say 'play the economist podcast', siri uses the music app to start playing the podcast. and lo-and-behold, you can see the chapters !
hope this helps. apple really need to fix this.
(iphone 5)