First Timothy and Intended Recipient

I’ve blogged about this a few times previously (here and here).

In working through the end of 1Ti 6, one comes across vv. 17-19. These are instructions to Timothy about those who are "rich in this present age":

17 As for the rich in this present age, charge them not to be haughty, nor to set their hopes on the uncertainty of riches, but on God, who richly provides us with everything to enjoy. 18 They are to do good, to be rich in good works, to be generous and ready to share, 19 thus storing up treasure for themselves as a good foundation for the future, so that they may take hold of that which is truly life. (1Ti 6.17-19, ESV)

If Paul is generally writing to the Ephesian community, why is he instructing Timothy to instruct those "rich in this present age"? If the letter is intended to be read to the community at large, wouldn’t these people be present at the reading?