The government can't fill every niche
Sure you'll have the basics: Food, building, entertainment, clothing etc which can be thought of and tracked by the government but then you'll have some entrepreneurs finding gaps in the market. A stronger material for protective boots, a new method of chocolate making, hand-made furniture from reclaimed wood...that sort of thing. These family businesses will all aim to fill some gap like that, they will have to be something new and better.
Now if I were the government I would, instead of ostracising these innovative businesses, have a level slightly higher to promote these businesses to. A government endorsed idea, the company gets some benefits and securities whilst giving up a little power. Their products can be made on government production lines, be spread throughout the country and the family can sit back a little.
You might be thinking that the government should just steal the idea and give the family no credit but this would give the families no incentive to create and every incentive not to give up their secrets. This way there is a form of pride in having made something the government wants - you are a family who has directly benefited the country. 207
Of course to most this is a pipe dream, a waste of time - time you could be giving to the country and community in other ways.
So you would get layers of business, government run ones, government endorsed ones (and probably layers within that) and purely family owned businesses.
Within the bottom two layers are competition. Family owned businesses will steal ideas from each other, try to promote themselves above competition to be noticed by the government - there would be no protection of their ideas (in fact if the government sees two companies with exactly the same idea they treat it as public property). This layer would be brutally competitive and, around this, would spring businesses in protection if your intellectual property, protection of your assets and businesses offering to infiltrate, spy and steal from your competitors.
The second layer would still be competitive, if your idea was close to another (perhaps someone created a stronger material for boots on building sites but someone else created a better rope) if you can improve your product to do the job of both then the other family will lose their status and your product will gain the resources which went into producing theirs. Again you will see, though slightly more elaborate due to the increased funds, espionage and protection businesses springing up. Of course this level has some degree of government protection, if you attack a government production line it is tantamount to treason. So the attacks are more tactical, the threats veiled and dealings with one another are polite. Inviting your worst enemy to dinner, serving them the finest food and exchanging small talk whilst you plan how best to discredit and steal from their family (or at least prevent them from doing so to you).
I could go on but I feel as though I'm going to end up writing a story if I do ^^; hope those ideas helped. 145