You should be absolutely fine with a narrow one, some people go crazy and use 36 or 34 cm bars. 40 or 38 is plenty if you ain't a broad guy.

Or even if you are. Some of the bigger riders in the pro peloton are using 38-cm bars for the aero advantage; a couple use 36 cm. But no one here is a pro racer, so use what feels comfortable.

Interesting report on the Cycling Weekly website.

Quoting an excerpt from the article (by writer/guinea pig Joe Baker; Mike Twelves was the aerodynamics engineer who ran the tests), where he discusses the most extreme position tested:

"Our final test runs gave us the opportunity to experiment with an extreme body position, using the narrow 30cm handlebar. We tested the aggressive drops position against an extreme ‘shrug’ position at 45kph and 55kph. The goal here was to show how a narrow bar can facilitate a more aerodynamic body position, albeit with some training - and it didn’t disappoint. At 45kph, the shrug position saved a further 24.7 watts compared to the aggressive drop position, and at 55kph, this grew to 42.6 watts. Compare that to the baseline handlebar, and you’ll find a nearly 70-watt saving.

"Twelves and I agreed that this was not a position you could hold for long periods of time, at least not without a lot of yoga and a very good chiropractor. Yet for short periods, it can be a lethal weapon in races."

(Warning: the writer semi-jokingly jeers at the idea of running 42-cm-wide handlebars on a racing bike in the first paragraph.)

How many watts can you really save running super narrow handlebars?

Why must one be a pro racer to want a faster, more efficient bike?

hard to say because it's a preferential thing. If you are mostly group riding and not racing just get whatever you feel most comfortable on and something that matches your shoulder width. I am a pretty small dude so I could do even a 36cm but very few if any manufactures makes one and most stock bikes in my size come with a 40cm, thus I either ride a stock 40cm or get a 38cm. To me the wider bars feel more stable and are better climbing bars since you can use them to leverage yourself. I just changed my carbon cockpit on my Scott Foil RC from a 40m to a 38cm and frankly I feel marginally more efficient but lighter in the wallet. This IMO is where a bike fit is super useful unless you know your component size from the get go. If you don't feel like dropping more money on marginally narrower bars I would work on bike positioning but also tilt your shifters inwards unless you are UCI racing. Also look into how your manufacture measures your bars. My ultralight Prime Primavera X-Light pro Bars feels narrower than my Syncros Creston iC SL 38cm bar and some of this is where/how a manufacture measures the width of the bar. I have heard of some ENVE bars marked as 38cm measuring smaller