By Keach Hagey and Amol Sharma 

Four of 10 dollars spent by U.S. pay-television operators on carriage fees would go to 21st Century Fox if its takeover bid for Time Warner Inc. succeeds, illustrating both the allure of the deal and its risks as the traditional cable-TV business model faces mounting pressures.

Combining 21st Century Fox's suite of channels, including FX and Fox News with Time Warner's TNT, TBS and HBO, would create a giant with 40% share of the estimated $40.3 billion U.S. cable and satellite operators spend on carriage fees, calculates MoffettNathanson analyst Michael Nathanson, more than double Fox's current 19% share.

The next biggest recipient is ESPN majority owner Walt Disney Co., which has a 25% market share, while Comcast Corp.'s NBCUniversal would follow with 10%.

Such sharply expanded presence in the cable channel market has obvious advantages for Fox. Fueled by subscription fees and advertising, cable channels have been the main source of revenue and profit growth for big U.S. entertainment companies over the past few years.

And programmers have been nervous that the consolidation among pay-TV companies, such as Comcast's pending bid for Time Warner Cable and AT&T's pending bid for DirecTV, would reduce their leverage in negotiating these fees unless they bulk up themselves.

"If I control more of the market, you are going to find it harder to live without me, and that way when I come to negotiate, I'm in a better position," said David Bank, an analyst at RBC Capital Markets.

On Wednesday, Time Warner confirmed it had rejected a takeover offer made by Fox last month valued at about $80 billion. Fox, however, hasn't given up the pursuit.

A successful acquisition would pose risks for Fox. The company would be doubling down on cable networks as the U.S. pay-TV market nears a saturation point. Nearly 90% of American households already subscribe to cable or satellite TV.

Pay-TV operators are struggling to find new customers. In some quarters in the past few years, the industry has contracted. There are fears that "cord cutting" will pick up or that young people will never sign up for cable TV when they become adults. Meanwhile, the very nature of TV is evolving quickly, as consumers look for on-demand content--often from streaming-video players like Netflix and Amazon. Some media executives believe traditional TV channels will have a smaller role in future years.

If it swallows Time Warner, 21st Century Fox would be far more exposed than any other entertainment company to these threats. That would put it under more pressure to look for growth in other ways.

While the combined company could continue to raise prices on distributors on a per-subscriber basis, that translates into higher cable-TV bills for consumers and isn't a long-term answer, executives at pay-TV distributors argue.

Another possibility: The combined company could be in a position to solve problems in the rollout of online availability of pay-TV services, through which subscribers can watch the channels that they pay for anywhere, either on iPads in the bedroom to smartphones on the bus, via mobile applications.

Programmers argue that the mobility of their channels--along with extra goodies baked into their apps, like past season shows and behind-the-scenes videos--add value to what customers get for their cable subscriptions. They have argued that such services justify higher per-subscriber affiliate fees.

Time Warner was an early advocate of this concept, dubbed "TV Everywhere, " which is designed to protect the pay-TV ecosystem from competition from less expensive online-video alternatives. HBO Go, the TV Everywhere app for HBO, is viewed as an industry leader.

But the effort has been slow to roll out across the industry, largely because every pay-TV operator must negotiate separate agreements with each cable channel to get the rights to put the shows online.

An additional complication is that channels generally don't control the digital rights to the shows they air, which are typically held by the production studio that made the show.

That means some channel apps often don't have access to all the shows that have aired on the channel--a phenomenon that hurts the channel's brand as more television viewing moves online.

Because both Fox and Time Warner own major film and television studios that produce many TV shows, a combination of the two companies could help their TV networks iron out these rights issues.

Content owners "want to ensure that rights issues are structured to protect the pay-TV ecosystem," wrote Mr. Nathanson in his research note. Time Warner and 21st Century Fox "have been aligned in this thinking and the combination would help drive a more unified industry standard" for putting content online and in cable video on demand, he wrote. The same enlarged presence in content production means that a combined Fox-Time Warner would be better positioned to survive a world in which channels matter less and content matters more.

"We are evolving into a world where Apple and Netflix are becoming a more important distribution platform," said Mr. Bank. "The more content you own, the more you are going to be able to set the agenda through the next half century for how content is paid for."

Write to Keach Hagey at keach.hagey@wsj.com and Amol Sharma at amol.sharma@wsj.com

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